Chapter 9 Algebraic Expressions and Identities (Additional Questions)

The correct option is (D).

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A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Let's examine each option:

(A) $x^2 + \frac{1}{x} = x^2 + x^{-1}$. The exponent of $x$ in the second term is $-1$, which is a negative integer. Thus, it is not a polynomial.

(B) $y^{3/2} - 5y$. The exponent of $y$ in the first term is $3/2$, which is a fraction (not an integer). Thus, it is not a polynomial.

(C) $3\sqrt{z} + 7 = 3z^{1/2} + 7$. The exponent of $z$ is $1/2$, which is a fraction (not an integer). Thus, it is not a polynomial.

(D) $p^4 - 3p^2 + 2p - 1$. The exponents of $p$ are $4$, $2$, and $1$ (since $2p = 2p^1$). The constant term $-1$ can be written as $-1 \cdot p^0$, where the exponent is $0$. All exponents ($4, 2, 1, 0$) are non-negative integers. The operations are subtraction and addition. Thus, this expression is a polynomial.

The correct option is (B).

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Terms in an algebraic expression are the parts of the expression that are added or subtracted. They include the sign that precedes them.

The given expression is $4x^2 - 7xy + 5y - 9$.

This expression can be written as the sum of its terms:

$4x^2 + (-7xy) + 5y + (-9)$

Therefore, the terms are $4x^2$, $-7xy$, $5y$, and $-9$.

The correct option is (B).

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In an algebraic expression, a coefficient is a numerical or literal factor of a term. The coefficient of a variable in a term is the factor by which the variable is multiplied.

The given expression is $5a^3 - 2a^2b + 7ab^2 - b^3$.

Let's identify the terms in the expression:

• Term 1: $5a^3$
• Term 2: $-2a^2b$
• Term 3: $7ab^2$
• Term 4: $-b^3$

We are asked to find the coefficient of $a^2$. We look for the term that contains $a^2$.

The term containing $a^2$ is $-2a^2b$.

In the term $-2a^2b$, $a^2$ is multiplied by $-2$ and $b$. So, the coefficient of $a^2$ in this term is $-2b$.

However, the options provided are numerical values. This indicates that the question is asking for the numerical coefficient of the term that includes $a^2$.

The numerical coefficient of the term $-2a^2b$ is -2.

The correct option is (A).

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Algebraic expressions are classified based on the number of terms they contain:

• An expression with one term is called a monomial.
• An expression with two terms is called a binomial.
• An expression with three terms is called a trinomial.
• An expression with one or more terms is generally called a polynomial.

The given expression is $3xy^2$. This expression consists of a single term.

Therefore, $3xy^2$ is a monomial.

The correct option is (A).

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To add the two expressions, we combine the like terms.

The given expressions are $(2x^2 + 3x - 5)$ and $(4x^2 - x + 7)$.

Adding them together:

$(2x^2 + 3x - 5) + (4x^2 - x + 7)$

Remove the parentheses:

$2x^2 + 3x - 5 + 4x^2 - x + 7$

Group the like terms:

$(2x^2 + 4x^2) + (3x - x) + (-5 + 7)$

Combine the coefficients of the like terms:

For the $x^2$ terms: $2x^2 + 4x^2 = (2+4)x^2 = 6x^2$

For the $x$ terms: $3x - x = 3x - 1x = (3-1)x = 2x$

For the constant terms: $-5 + 7 = 2$

Putting it all together:

$(2x^2 + 4x^2) + (3x - x) + (-5 + 7) = 6x^2 + 2x + 2$

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The sum of the expressions is $6x^2 + 2x + 2$.

The correct option is (A).

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To subtract the first expression from the second, we write:

$(3y^2 + y - 4) - (5y^2 - 2y + 1)$

When subtracting a polynomial, we change the sign of each term in the polynomial being subtracted and then add the resulting expression.

$(3y^2 + y - 4) + (- (5y^2) - (-2y) - (1))$

$3y^2 + y - 4 - 5y^2 + 2y - 1$

Now, group the like terms:

$(3y^2 - 5y^2) + (y + 2y) + (-4 - 1)$

Combine the coefficients of the like terms:

For the $y^2$ terms: $3y^2 - 5y^2 = (3-5)y^2 = -2y^2$

For the $y$ terms: $y + 2y = 1y + 2y = (1+2)y = 3y$

For the constant terms: $-4 - 1 = -5$

Putting it all together:

$(3y^2 - 5y^2) + (y + 2y) + (-4 - 1) = -2y^2 + 3y - 5$

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The result of the subtraction is $-2y^2 + 3y - 5$.

The correct option is (C).

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To multiply two monomials, we multiply their numerical coefficients and the variable parts separately.

We need to multiply $6x$ by $(-3y)$.

Multiply the coefficients: $6 \times (-3) = -18$.

Multiply the variables: $x \times y = xy$.

Combine the results:

$(6x) \times (-3y) = (6 \times -3) \times (x \times y)$

$(6x) \times (-3y) = -18xy$

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The product is $-18xy$.

The correct option is (A).

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To find the product of two binomials, we use the distributive property. We multiply each term in the first binomial by each term in the second binomial and then combine the like terms.

We need to find the product of $(2x + 3)(x - 5)$.

Using the distributive property (or FOIL method):

Multiply the First terms: $(2x)(x) = 2x^2$

Multiply the Outer terms: $(2x)(-5) = -10x$

Multiply the Inner terms: $(3)(x) = 3x$

Multiply the Last terms: $(3)(-5) = -15$

Add these products together:

$(2x + 3)(x - 5) = 2x^2 + (-10x) + 3x + (-15)$

$(2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15$

Now, combine the like terms (the terms with $x$):

$-10x + 3x = (-10 + 3)x = -7x$

So, the expression becomes:

$2x^2 - 7x - 15$

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Comparing this result with the given options, we see that option (A) shows the correct intermediate step and the correct final product.

$(2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15$

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Note that option (C) also shows a correct intermediate step and the same correct final product, just with the order of the middle terms swapped.

The correct option is (A).

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We can use the algebraic identity for the square of a sum:

$(x + y)^2 = x^2 + 2xy + y^2$

In the given expression $(a + 5)^2$, we have $x = a$ and $y = 5$.

Substituting these values into the identity:

$(a + 5)^2 = (a)^2 + 2(a)(5) + (5)^2$

$(a + 5)^2 = a^2 + 10a + 25$

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Alternatively, we can expand by multiplication:

$(a + 5)^2 = (a + 5)(a + 5)$

Using the distributive property:

$(a + 5)(a + 5) = a(a+5) + 5(a+5)$

$= a \times a + a \times 5 + 5 \times a + 5 \times 5$

$= a^2 + 5a + 5a + 25$

Combine the like terms ($5a + 5a = 10a$):

$= a^2 + 10a + 25$

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The expansion of $(a + 5)^2$ is $a^2 + 10a + 25$.

The correct option is (C).

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To simplify the expression $(3x - 2y)^2$, we can use the algebraic identity for the square of a difference:

$(a - b)^2 = a^2 - 2ab + b^2$

In this expression, we have $a = 3x$ and $b = 2y$.

Substitute these values into the identity:

$(3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2$

Calculate each term:

$(3x)^2 = 3^2 \times x^2 = 9x^2$

$2(3x)(2y) = 2 \times 3 \times 2 \times x \times y = 12xy$

$(2y)^2 = 2^2 \times y^2 = 4y^2$

Substitute these back into the expansion:

$(3x - 2y)^2 = 9x^2 - 12xy + 4y^2$

This can also be written as $9x^2 + 4y^2 - 12xy$.

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The simplified expression is $9x^2 + 4y^2 - 12xy$.

The correct option is (B).

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We need to find the product of $(x + 7)(x - 7)$.

This expression is in the form $(a + b)(a - b)$, where $a = x$ and $b = 7$.

We can use the algebraic identity for the difference of squares:

$(a + b)(a - b) = a^2 - b^2$

Substitute $a = x$ and $b = 7$ into the identity:

$(x + 7)(x - 7) = (x)^2 - (7)^2$

$(x + 7)(x - 7) = x^2 - 49$

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Alternatively, we can expand by using the distributive property (FOIL):

$(x + 7)(x - 7) = x(x - 7) + 7(x - 7)$

$= x \times x + x \times (-7) + 7 \times x + 7 \times (-7)$

$= x^2 - 7x + 7x - 49$

Combine the like terms ($-7x + 7x = 0$):

$= x^2 + 0x - 49$

$= x^2 - 49$

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The value of $(x + 7)(x - 7)$ is $x^2 - 49$.

The correct option is (A).

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We need to find the value of $103 \times 97$ using a suitable identity.

We can rewrite the numbers $103$ and $97$ in terms of a common base, which is $100$ in this case:

$103 = 100 + 3$

$97 = 100 - 3$

So, the product $103 \times 97$ can be written as $(100 + 3)(100 - 3)$.

This expression is in the form $(a + b)(a - b)$, which is the algebraic identity for the difference of squares:

Comparing $(100 + 3)(100 - 3)$ with $(a + b)(a - b)$, we have $a = 100$ and $b = 3$.

Applying the identity (1):

$(100 + 3)(100 - 3) = (100)^2 - (3)^2$

Calculate the squares:

$(100)^2 = 100 \times 100 = 10000$

$(3)^2 = 3 \times 3 = 9$

Substitute these values back into the expression:

$(100 + 3)(100 - 3) = 10000 - 9$

Subtracting the numbers:

Thus, $103 \times 97 = 9991$.

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Option (A) correctly uses the identity $(100+3)(100-3) = 100^2 - 3^2$ and arrives at the correct result $10000 - 9 = 9991$.

Option (B) attempts to use numbers that don't represent the original product.

Option (C) attempts a different approach but doesn't use the most suitable identity for the product of $103$ and $97$.

Option (D) uses the distributive property, which is a valid calculation method, but the question asks to use a "suitable identity," and the difference of squares is the most direct identity for $(100+3)(100-3)$.

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The value of $103 \times 97$ using the difference of squares identity is 9991.

The correct option is (A).

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To divide a monomial by another monomial, we divide the numerical coefficients and then divide the variable parts using the rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

We are asked to divide $12x^3y^2$ by $4xy$. We can write this as a fraction:

$\frac{12x^3y^2}{4xy}$

Now, divide the coefficients and the variables separately:

Divide the numerical coefficients:

$\frac{12}{4} = 3$

Divide the variable $x$ terms:

$\frac{x^3}{x} = \frac{x^3}{x^1} = x^{3-1} = x^2$

Divide the variable $y$ terms:

$\frac{y^2}{y} = \frac{y^2}{y^1} = y^{2-1} = y^1 = y$

Multiply the results together:

$3 \times x^2 \times y = 3x^2y$

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Alternatively, we can show the cancellation of common factors:

$\frac{\cancel{12}^{3} \times x^3 \times y^2}{\cancel{4}_{1} \times x \times y} = \frac{3 \times x^{3-1} \times y^{2-1}}{1} = 3x^2y$

Using cancellation marks as requested:

$\frac{\cancel{12}^{3} x^{\cancel{3}2} y^{\cancel{2}1}}{\cancel{4}_{1} \cancel{x}_{1} \cancel{y}_{1}} = 3x^2y$

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The result of the division is $3x^2y$.

The correct options are (A), (C), and (E).

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Like terms are terms that have the same variables raised to the same powers. The numerical coefficient does not affect whether terms are like terms.

The given term is $3x^2y$. The variables and their powers are $x^2$ and $y^1$.

Let's examine each option:

(A) $-5x^2y$: This term has the variables $x$ raised to the power of $2$ ($x^2$) and $y$ raised to the power of $1$ ($y^1$). These match the variables and powers in $3x^2y$. Therefore, $-5x^2y$ is a like term with $3x^2y$.

(B) $7xy^2$: This term has the variable $x$ raised to the power of $1$ ($x^1$) and $y$ raised to the power of $2$ ($y^2$). The powers of $x$ and $y$ do not match those in $3x^2y$. Therefore, $7xy^2$ is not a like term with $3x^2y$.

(C) $y x^2$: This term has the variable $y$ raised to the power of $1$ ($y^1$) and $x$ raised to the power of $2$ ($x^2$). The order of variables does not matter in multiplication ($yx^2$ is the same as $x^2y$). These match the variables and powers in $3x^2y$. Therefore, $y x^2$ is a like term with $3x^2y$.

(D) $2x^2$: This term has the variable $x$ raised to the power of $2$ ($x^2$) but does not contain the variable $y$ (or $y^0$). The variables do not match those in $3x^2y$. Therefore, $2x^2$ is not a like term with $3x^2y$.

(E) $-8yx^2$: This term has the variable $y$ raised to the power of $1$ ($y^1$) and $x$ raised to the power of $2$ ($x^2$). The order of variables does not matter ($yx^2$ is the same as $x^2y$). These match the variables and powers in $3x^2y$. Therefore, $-8yx^2$ is a like term with $3x^2y$.

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Terms with the same variables raised to the same powers are like terms. Options (A), (C), and (E) satisfy this condition compared to the term $3x^2y$.

The correct option is (D).

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Let's evaluate the Assertion and the Reason.

Assertion (A): The expression $2x^2 + 3\sqrt{x} - 5$ is a polynomial.

Recall that $\sqrt{x}$ can be written in exponential form as $x^{1/2}$.

So, the expression is $2x^2 + 3x^{1/2} - 5$.

A polynomial is an algebraic expression where the powers of the variables are non-negative integers ($0, 1, 2, 3, ...$).

In the term $3x^{1/2}$, the power of the variable $x$ is $1/2$, which is a fraction and not an integer.

Therefore, the expression $2x^2 + 3\sqrt{x} - 5$ is not a polynomial.

Thus, Assertion (A) is false.

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Reason (R): A polynomial is an algebraic expression where the powers of the variables are non-negative integers.

This statement is the correct definition of a polynomial.

Thus, Reason (R) is true.

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Since Assertion (A) is false and Reason (R) is true, the correct option is (D).

The correct option is (A).

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Let's classify each example expression based on the number of terms:

(a) $a^2 + b^2 + c^2$: This expression has three terms ($a^2$, $b^2$, and $c^2$). It is a trinomial.

(b) $pqr$: This expression has one term ($pqr$). It is a monomial.

(c) $x+y+z+w$: This expression has four terms ($x$, $y$, $z$, and $w$). An expression with four or more terms is generally referred to as a polynomial.

(d) $m-n$: This expression has two terms ($m$ and $-n$). It is a binomial.

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Now, let's match the expression types with their examples:

• (i) Monomial matches with (b) $pqr$.
• (ii) Binomial matches with (d) $m-n$.
• (iii) Trinomial matches with (a) $a^2 + b^2 + c^2$.
• (iv) Polynomial (general) matches with (c) $x+y+z+w$ (as it's the example with more than three terms).

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The correct matching is: (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c).

Comparing this with the given options, we find that option (A) matches this arrangement.

The correct option is (B).

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We need to simplify the expression $(a + b)^2 - (a - b)^2$.

We can use the algebraic identities for the square of a sum and the square of a difference:

Now, subtract the second identity (2) from the first identity (1):

$(a + b)^2 - (a - b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)$

Remove the parentheses, remembering to change the sign of each term in the second polynomial:

$(a + b)^2 - (a - b)^2 = a^2 + 2ab + b^2 - a^2 + 2ab - b^2$

Group the like terms:

$(a^2 - a^2) + (2ab + 2ab) + (b^2 - b^2)$

Combine the like terms:

$a^2 - a^2 = 0$

$2ab + 2ab = 4ab$

$b^2 - b^2 = 0$

So, the simplified expression is:

$(a + b)^2 - (a - b)^2 = 0 + 4ab + 0$

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Alternatively, we can treat this as a difference of squares itself, where $A = (a+b)$ and $B = (a-b)$.

Using the identity $A^2 - B^2 = (A+B)(A-B)$:

$(a + b)^2 - (a - b)^2 = ((a + b) + (a - b))((a + b) - (a - b))$

Simplify the terms inside the parentheses:

$((a + b) + (a - b)) = a + b + a - b = (a+a) + (b-b) = 2a + 0 = 2a$

$((a + b) - (a - b)) = a + b - a + b = (a-a) + (b+b) = 0 + 2b = 2b$

Now, multiply the simplified terms:

$(a + b)^2 - (a - b)^2 = (2a)(2b)$

$(a + b)^2 - (a - b)^2 = 4ab$

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The simplified expression is $4ab$.

The correct option is (B).

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The length of the rectangular park is given by the expression $(2x + 3)$ meters.

This expression has two terms: $2x$ and $3$.

The width of the rectangular park is given by the expression $(x - 1)$ meters.

This expression has two terms: $x$ and $-1$.

An algebraic expression with exactly two terms is classified as a binomial.

Since both the length $(2x + 3)$ and the width $(x - 1)$ are algebraic expressions containing two terms, they are both binomials.

The correct option is (A).

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From Question 18, the length of the rectangular park is $(2x + 3)$ meters and the width is $(x - 1)$ meters.

The area of a rectangle is given by the formula:

Area = Length $\times$ Width

Substitute the given expressions for length and width:

Area $= (2x + 3)(x - 1)$

To find the expression for the area, we need to multiply the two binomials. We can use the distributive property (FOIL method):

Multiply the First terms: $(2x)(x) = 2x^2$

Multiply the Outer terms: $(2x)(-1) = -2x$

Multiply the Inner terms: $(3)(x) = 3x$

Multiply the Last terms: $(3)(-1) = -3$

Add these products together:

Area $= 2x^2 + (-2x) + 3x + (-3)$

Area $= 2x^2 - 2x + 3x - 3$

Combine the like terms (the terms with $x$):

$-2x + 3x = (-2 + 3)x = 1x = x$

So, the expression for the area becomes:

Area $= 2x^2 + x - 3$

The unit of area is square meters.

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Comparing this result with the given options, option (A) shows the correct intermediate step and the correct final expression for the area.

$(2x + 3)(x - 1) = 2x^2 - 2x + 3x - 3 = 2x^2 + x - 3$ sq meters

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The expression for the area of the park is $2x^2 + x - 3$ sq meters.

The correct option is (D).

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From Question 18, the length of the park is $(2x + 3)$ meters and the width is $(x - 1)$ meters.

From Question 19, the expression for the area of the park is $2x^2 + x - 3$ square meters.

We are given that $x = 5$ meters.

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Method 1: Calculate length and width first, then area.

Length $= 2x + 3 = 2(5) + 3 = 10 + 3 = 13$ meters.

Width $= x - 1 = 5 - 1 = 4$ meters.

Area = Length $\times$ Width $= 13 \times 4 = 52$ square meters.

Option (A) demonstrates this method and calculation correctly: $(2 \times 5 + 3)(5 - 1) = (13)(4) = 52$.

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Method 2: Substitute $x=5$ into the area expression.

Area $= 2x^2 + x - 3$

Substitute $x=5$:

Area $= 2(5)^2 + (5) - 3$

Area $= 2(25) + 5 - 3$

Area $= 50 + 5 - 3$

Area $= 55 - 3$

Area $= 52$ square meters.

Option (B) demonstrates this method and calculation correctly: $2(5^2) + 5 - 3 = 2(25) + 5 - 3 = 50 + 5 - 3 = 52$. Note that the intermediate step $50+2$ shown in option (B) is correct as $5-3=2$.

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Both methods yield the same result, which is 52 square meters.

Option (A) is a correct calculation method and result.

Option (B) is a correct calculation method and result.

Option (C) states the correct final area.

Since options (A), (B), and (C) are all correct based on the problem, the answer is All of the above.

The correct option is (B).

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To divide the polynomial $(6x^2 + 9x)$ by the monomial $3x$, we can divide each term of the polynomial by the monomial separately.

We need to calculate $\frac{6x^2 + 9x}{3x}$.

We can write this as:

$\frac{6x^2}{3x} + \frac{9x}{3x}$

Now, simplify each fraction:

For the first term, $\frac{6x^2}{3x}$:

Divide the coefficients: $\frac{6}{3} = 2$.

Divide the variable $x$ terms: $\frac{x^2}{x} = x^{2-1} = x^1 = x$.

So, $\frac{6x^2}{3x} = 2x$.

For the second term, $\frac{9x}{3x}$:

Divide the coefficients: $\frac{9}{3} = 3$.

Divide the variable $x$ terms: $\frac{x}{x} = x^{1-1} = x^0$. Note that any non-zero number raised to the power of $0$ is $1$ ($x^0 = 1$ for $x \neq 0$).

So, $\frac{9x}{3x} = 3 \times 1 = 3$.

Now, add the results of the two divisions:

$\frac{6x^2 + 9x}{3x} = 2x + 3$

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Alternatively, we can factor the numerator and then cancel common factors:

The common factor in $6x^2$ and $9x$ is $3x$.

$6x^2 + 9x = 3x(2x) + 3x(3) = 3x(2x + 3)$

Now substitute this back into the division:

$\frac{3x(2x + 3)}{3x}$

Cancel the common factor $3x$ in the numerator and the denominator (assuming $x \neq 0$):

$\frac{\cancel{3x}(2x + 3)}{\cancel{3x}} = 2x + 3$

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The result of the division is $2x + 3$.

The correct option is (D).

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We need to identify which of the given algebraic identities is incorrect.

Let's examine each identity:

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(A) $(a+b)^2 = a^2 + 2ab + b^2$

This is the standard identity for the square of a sum. It is correct.

We can verify this by expanding $(a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$.

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(B) $(a-b)^2 = a^2 - 2ab + b^2$

This is the standard identity for the square of a difference. It is correct.

We can verify this by expanding $(a-b)^2 = (a-b)(a-b) = a(a-b) - b(a-b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2$.

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(C) $(a+b)(a-b) = a^2 - b^2$

This is the standard identity for the difference of squares. It is correct.

We can verify this by expanding $(a+b)(a-b) = a(a-b) + b(a-b) = a^2 - ab + ba - b^2 = a^2 - b^2$.

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(D) $(x+a)(x+b) = x^2 + (a+b)x + ab^2$

Let's expand the left side of the equation:

$(x+a)(x+b) = x(x+b) + a(x+b)$

$= x \times x + x \times b + a \times x + a \times b$

$= x^2 + bx + ax + ab$

Combine the terms with $x$:

$= x^2 + (b+a)x + ab$

$= x^2 + (a+b)x + ab$

The correct identity is $(x+a)(x+b) = x^2 + (a+b)x + ab$.

The given identity is $(x+a)(x+b) = x^2 + (a+b)x + ab^2$. The last term is $ab^2$ instead of $ab$.

Therefore, this identity is incorrect.

The correct option is (A).

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To simplify the expression $(p + q + r)^2$, we can use the algebraic identity for the square of a trinomial:

In the given expression $(p + q + r)^2$, we have $a = p$, $b = q$, and $c = r$.

Substitute these values into the identity (1):

$(p + q + r)^2 = (p)^2 + (q)^2 + (r)^2 + 2(p)(q) + 2(q)(r) + 2(r)(p)$

$(p + q + r)^2 = p^2 + q^2 + r^2 + 2pq + 2qr + 2rp$

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Alternatively, we can expand the expression by multiplication:

$(p + q + r)^2 = (p + q + r)(p + q + r)$

Distribute each term from the first factor to the second factor:

$= p(p + q + r) + q(p + q + r) + r(p + q + r)$

$= (p \times p + p \times q + p \times r) + (q \times p + q \times q + q \times r) + (r \times p + r \times q + r \times r)$

$= (p^2 + pq + pr) + (qp + q^2 + qr) + (rp + rq + r^2)$

Combine the terms:

$= p^2 + q^2 + r^2 + pq + pr + qp + qr + rp + rq$

Group and combine the like terms (remembering $pq = qp$, $pr = rp$, $qr = rq$):

$= p^2 + q^2 + r^2 + (pq + qp) + (pr + rp) + (qr + rq)$

$= p^2 + q^2 + r^2 + 2pq + 2pr + 2qr$

Rearranging the cross-product terms in the order $pq, qr, rp$:

$= p^2 + q^2 + r^2 + 2pq + 2qr + 2rp$

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The simplified expression is $p^2 + q^2 + r^2 + 2pq + 2qr + 2rp$.

Comparing this result with the given options, option (A) matches the expanded form.

Note that option (D) $p^2 + q^2 + r^2 + 2(pq + qr + rp)$ is also mathematically equivalent to the correct expansion, as $2(pq + qr + rp) = 2pq + 2qr + 2rp$. However, option (A) represents the fully expanded form, which is typically what is expected in such questions unless factorization is explicitly requested for part of the expression.

The correct option is (C).

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The degree of a polynomial with multiple variables is the highest sum of the exponents of the variables in any single term of the polynomial.

The given polynomial is $7x^3y^2 + 5xy^4 - 9x^2y^2$.

We need to find the degree of each term:

• Term 1: $7x^3y^2$. The variables are $x$ and $y$. The exponent of $x$ is $3$ and the exponent of $y$ is $2$. The sum of the exponents is $3 + 2 = 5$. So, the degree of this term is 5.
• Term 2: $5xy^4$. The variables are $x$ and $y$. The exponent of $x$ is $1$ (since $xy^4 = x^1y^4$) and the exponent of $y$ is $4$. The sum of the exponents is $1 + 4 = 5$. So, the degree of this term is 5.
• Term 3: $-9x^2y^2$. The variables are $x$ and $y$. The exponent of $x$ is $2$ and the exponent of $y$ is $2$. The sum of the exponents is $2 + 2 = 4$. So, the degree of this term is 4.

The degrees of the terms are 5, 5, and 4.

The degree of the polynomial is the highest of these degrees.

Highest degree = $\max(5, 5, 4) = 5$.

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Therefore, the degree of the polynomial $7x^3y^2 + 5xy^4 - 9x^2y^2$ is 5.

The correct option is (A).

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Let the expression that should be added to $x^2 + xy + y^2$ be $P$.

According to the problem statement, we have:

To find the expression $P$, we need to subtract the polynomial $(x^2 + xy + y^2)$ from the polynomial $(2x^2 + 3xy)$.

So, $P = (2x^2 + 3xy) - (x^2 + xy + y^2)$

To subtract the polynomial in the parentheses, we change the sign of each term inside the parentheses and then add the resulting expression:

$P = 2x^2 + 3xy - x^2 - xy - y^2$

Now, group the like terms together:

$P = (2x^2 - x^2) + (3xy - xy) + (-y^2)$

Combine the like terms:

$P = (2-1)x^2 + (3-1)xy - y^2$

$P = 1x^2 + 2xy - y^2$

$P = x^2 + 2xy - y^2$

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Thus, the expression that should be added is $x^2 + 2xy - y^2$.

Comparing this result with the given options, we find that it matches option (A).

The expression to be added is $x^2 + 2xy - y^2$.

The correct option is (A).

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We need to simplify the expression $(x+3)(x+4) - (x+2)(x+5)$.

First, let's expand the first product $(x+3)(x+4)$ using the identity $(x+a)(x+b) = x^2 + (a+b)x + ab$ with $a=3$ and $b=4$:

$(x+3)(x+4) = x^2 + (3+4)x + (3)(4)$

Next, let's expand the second product $(x+2)(x+5)$ using the same identity with $a=2$ and $b=5$:

$(x+2)(x+5) = x^2 + (2+5)x + (2)(5)$

Now, subtract the second expanded expression (2) from the first expanded expression (1):

$(x^2 + 7x + 12) - (x^2 + 7x + 10)$

Remove the parentheses, remembering to change the sign of each term in the second polynomial:

$= x^2 + 7x + 12 - x^2 - 7x - 10$

Group the like terms:

$= (x^2 - x^2) + (7x - 7x) + (12 - 10)$

Combine the like terms:

$= 0x^2 + 0x + 2$

$= 2$

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The simplified expression is 2.

Option (A) correctly shows the expansion of both products and the final result of the subtraction.

The correct option is (B).

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The sentence is "An expression containing only one term is called a _________."

Based on the classification of algebraic expressions by the number of terms:

• A monomial is an expression that contains only one term.
• A binomial is an expression that contains two terms.
• A trinomial is an expression that contains three terms.
• A polynomial is a general term for an expression containing one or more terms, where the exponents of the variables are non-negative integers. Monomials, binomials, and trinomials are all types of polynomials.

The blank in the sentence should be filled with the term for an expression with only one term.

Therefore, the correct word is Monomial.

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The completed sentence is: An expression containing only one term is called a Monomial.

The correct option is (A).

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A monomial is an algebraic expression consisting of only one term. A term is a product of coefficients and variables raised to non-negative integer powers.

Let's consider two generic monomials:

Monomial 1: $Ax^ay^bz^c...$ (where $A$ is the coefficient and $a, b, c, ...$ are non-negative integers)

Monomial 2: $Bx^dy^ez^f...$ (where $B$ is the coefficient and $d, e, f, ...$ are non-negative integers)

The product of these two monomials is:

$(Ax^ay^bz^c...) \times (Bx^dy^ez^f...) = (A \times B) \times (x^a \times x^d) \times (y^b \times y^e) \times (z^c \times z^f)...$

Using the rule for multiplying exponents with the same base ($\left.a^m \times a^n = a^{m+n}\right.$):

Product $= (AB) \times x^{a+d} \times y^{b+e} \times z^{c+f}...$

Let $C = AB$, and the new exponents be $m = a+d$, $n = b+e$, $p = c+f$, etc.

Product $= Cx^my^nz^p...$

Since $A$ and $B$ are coefficients, their product $C$ is also a coefficient.

Since $a, b, c, ..., d, e, f, ...$ are non-negative integers, their sums $m = a+d$, $n = b+e$, $p = c+f$, etc., are also non-negative integers.

The resulting expression $Cx^my^nz^p...$ consists of a single term, which is a product of a coefficient and variables raised to non-negative integer powers.

This matches the definition of a monomial.

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Example:

Let Monomial 1 be $3x^2y$ and Monomial 2 be $-2xy^3$.

Product $= (3x^2y) \times (-2xy^3)$

$= (3 \times -2) \times (x^2 \times x^1) \times (y^1 \times y^3)$

$= -6 \times x^{2+1} \times y^{1+3}$

$= -6x^3y^4$

$-6x^3y^4$ is a single term with variables raised to non-negative integer powers. Thus, it is a monomial.

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Therefore, the product of two monomials is always a Monomial.

The correct option is (D).

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We need to find the value of $10.5^2$ using a suitable identity. Let's examine each option as presented, as they show different calculation methods.

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Option (A):

$(10+0.5)^2 = 10^2 + 2(10)(0.5) + 0.5^2$

This uses the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $a=10$ and $b=0.5$.

Calculating the terms:

$10^2 = 100$

$2(10)(0.5) = 20 \times 0.5 = 10$

$0.5^2 = 0.5 \times 0.5 = 0.25$

Summing the terms: $100 + 10 + 0.25 = 110.25$.

This calculation is correct and uses an identity.

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Option (B):

$(10.5)^2 = 10.5 \times 10.5 = 110.25$

This is a direct multiplication calculation. While it doesn't use a specific algebraic identity in the same way as (A) or (C), it is a correct method to calculate the square and the result is correct.

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Option (C):

$(11-0.5)^2 = 11^2 - 2(11)(0.5) + 0.5^2$

This uses the identity $(a-b)^2 = a^2 - 2ab + b^2$ with $a=11$ and $b=0.5$.

Calculating the terms:

$11^2 = 121$

$2(11)(0.5) = 22 \times 0.5 = 11$

$0.5^2 = 0.25$

Calculating the difference: $121 - 11 + 0.25 = 110 + 0.25 = 110.25$.

This calculation is correct and uses an identity.

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All three options demonstrate calculations that result in the correct value for $10.5^2$, which is 110.25. Options (A) and (C) specifically use algebraic identities as requested in the question prompt.

Option (D) states that "All of the above are correct calculations." Since the calculations shown in (A), (B), and (C) all correctly arrive at the value 110.25, option (D) is true.

The value of $10.5^2$ is 110.25.

The correct option is (A).

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To divide a polynomial by a monomial, we divide each term of the polynomial by the monomial separately.

We need to calculate $\frac{15x^3 - 9x^2}{3x^2}$.

We can rewrite this expression as the difference of two fractions:

$\frac{15x^3}{3x^2} - \frac{9x^2}{3x^2}$

Now, let's simplify each fraction:

For the first term, $\frac{15x^3}{3x^2}$:

Divide the coefficients: $\frac{15}{3} = 5$.

Divide the variable terms using the exponent rule $\frac{a^m}{a^n} = a^{m-n}$: $\frac{x^3}{x^2} = x^{3-2} = x^1 = x$.

So, the first term simplifies to $5x$.

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For the second term, $\frac{9x^2}{3x^2}$:

Divide the coefficients: $\frac{9}{3} = 3$.

Divide the variable terms: $\frac{x^2}{x^2} = x^{2-2} = x^0$. Assuming $x \neq 0$, any non-zero term raised to the power of $0$ is $1$. So, $x^0 = 1$.

So, the second term simplifies to $3 \times 1 = 3$.

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Now, combine the simplified terms with the subtraction sign:

$\frac{15x^3}{3x^2} - \frac{9x^2}{3x^2} = 5x - 3$

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The result of dividing $(15x^3 - 9x^2)$ by $3x^2$ is $5x - 3$.

The correct option is (D).

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Let's evaluate the Assertion and the Reason.

Assertion (A): $(x+y)^2 = x^2 + y^2$ is an algebraic identity.

An algebraic identity is an equality that holds true for all values of the variables involved.

Let's expand the left side of the equation using the correct identity for the square of a sum:

Comparing the expansion $(x^2 + 2xy + y^2)$ with the right side of the given assertion $(x^2 + y^2)$, we see that the term $2xy$ is missing on the right side of the assertion.

The equality $(x+y)^2 = x^2 + y^2$ holds true only if $2xy = 0$, which means either $x=0$ or $y=0$. It does not hold true for all values of $x$ and $y$ (for example, if $x=1$ and $y=1$, the left side is $(1+1)^2 = 2^2 = 4$, but the right side is $1^2 + 1^2 = 1+1=2$; $4 \neq 2$).

Therefore, the equality $(x+y)^2 = x^2 + y^2$ is not an identity.

Thus, Assertion (A) is false.

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Reason (R): An identity is an equality that holds true for all values of the variables.

This is the precise definition of an algebraic identity.

Thus, Reason (R) is true.

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Since Assertion (A) is false and Reason (R) is true, the correct option is (D).

The correct option is (B).

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We need to find the expression equivalent to $(5a - 3b)(5a + 3b)$.

This product is in the form of the algebraic identity for the difference of squares:

Comparing $(5a - 3b)(5a + 3b)$ with $(x - y)(x + y)$, we can see that $x = 5a$ and $y = 3b$.

Substitute these values into the identity (1):

$(5a - 3b)(5a + 3b) = (5a)^2 - (3b)^2$

Calculate the square of each term:

$(5a)^2 = 5^2 \times a^2 = 25a^2$

$(3b)^2 = 3^2 \times b^2 = 9b^2$

Substitute these results back into the expression:

$(5a - 3b)(5a + 3b) = 25a^2 - 9b^2$

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The expression equivalent to $(5a - 3b)(5a + 3b)$ is $25a^2 - 9b^2$.

The correct option is (D).

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The original square garden has a side length given by the expression $(3y + 2)$ meters.

The area of a square is given by the formula: Area = (side length)$^2$.

So, the area of the original square garden is $(3y + 2)^2$ square meters.

To find the expression for the area, we need to expand $(3y + 2)^2$. We can use the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $a = 3y$ and $b = 2$.

Area $= (3y + 2)^2 = (3y)^2 + 2(3y)(2) + (2)^2$

Area $= 9y^2 + 12y + 4$ square meters.

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Let's examine the given options:

(A) $(3y+2)^2 = 9y^2 + 12y + 4$ sq meters. This option correctly uses the identity and calculates the area.

(B) $9y^2 + 4$ sq meters. This option incorrectly calculates $(3y+2)^2$ by squaring each term separately, omitting the middle term $2(3y)(2)$.

(C) $(3y+2)(3y+2) = 9y^2 + 6y + 6y + 4 = 9y^2 + 12y + 4$ sq meters. This option correctly expands $(3y+2)^2$ by multiplying the binomials directly and then combining like terms. The intermediate steps and the final result are correct.

(D) Both (A) and (C). Since both option (A) and option (C) correctly calculate the area of the original square garden, this option is correct.

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The area of the original square garden is $9y^2 + 12y + 4$ sq meters.

The correct option is (D).

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From the case study description in Question 33, the original side length is $(3y+2)$ meters. The owner increases the side length by 1 meter.

The new side length is: $(3y+2) + 1 = 3y+3$ meters.

The new garden is also a square with side length $(3y+3)$ meters.

The area of a square is given by the formula: Area = (side length)$^2$.

So, the area of the new enlarged square garden is $(3y + 3)^2$ square meters.

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To find the expression for the area, we need to expand $(3y + 3)^2$.

Method 1: Using the identity $(a+b)^2 = a^2 + 2ab + b^2$.

Here, $a = 3y$ and $b = 3$.

Area $= (3y + 3)^2 = (3y)^2 + 2(3y)(3) + (3)^2$

Area $= 9y^2 + 18y + 9$ square meters.

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Method 2: Expanding by multiplication (FOIL).

Area $= (3y + 3)(3y + 3)$

Multiply the First terms: $(3y)(3y) = 9y^2$

Multiply the Outer terms: $(3y)(3) = 9y$

Multiply the Inner terms: $(3)(3y) = 9y$

Multiply the Last terms: $(3)(3) = 9$

Add these products together:

Area $= 9y^2 + 9y + 9y + 9$

Combine the like terms ($9y + 9y = 18y$):

Area $= 9y^2 + 18y + 9$ square meters.

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Let's examine the given options:

(A) $(3y+3)^2 = 9y^2 + 18y + 9$ sq meters. This option correctly uses the identity and shows the correct expanded form and result.

(B) $9y^2 + 9$ sq meters. This is an incorrect expansion of $(3y+3)^2$, missing the middle term $2(3y)(3) = 18y$.

(C) $(3y+3)(3y+3) = 9y^2 + 9y + 9y + 9 = 9y^2 + 18y + 9$ sq meters. This option correctly expands the square by direct multiplication and shows the intermediate steps and the final result, which is correct.

(D) Both (A) and (C). Since both option (A) and option (C) correctly calculate and express the area of the new enlarged square garden, this option is correct.

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The area of the new enlarged square garden is $9y^2 + 18y + 9$ sq meters.

The correct option is (A).

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From Question 33, the area of the original square garden is $9y^2 + 12y + 4$ square meters.

From Question 34, the area of the new enlarged square garden is $9y^2 + 18y + 9$ square meters.

The increase in the area is the difference between the new area and the original area.

Increase in Area = Area of New Garden - Area of Original Garden

Increase in Area $= (9y^2 + 18y + 9) - (9y^2 + 12y + 4)$

To perform the subtraction, we change the sign of each term in the second polynomial and add:

Increase in Area $= 9y^2 + 18y + 9 - 9y^2 - 12y - 4$

Group the like terms:

Increase in Area $= (9y^2 - 9y^2) + (18y - 12y) + (9 - 4)$

Combine the like terms:

$9y^2 - 9y^2 = 0$

$18y - 12y = (18 - 12)y = 6y$

$9 - 4 = 5$

So, the increase in area is:

Increase in Area $= 0 + 6y + 5$

Increase in Area $= 6y + 5$ square meters.

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Option (A) correctly shows the subtraction of the two area expressions and arrives at the correct result $6y + 5$.

The increase in the area of the garden is $6y + 5$ sq meters.

The correct option is (B).

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We need to divide the polynomial $(p^2 - 8p + 15)$ by the polynomial $(p - 3)$. We can use polynomial long division.

Set up the long division:

Steps for long division:

1. Divide the first term of the dividend ($p^2$) by the first term of the divisor ($p$). $p^2 / p = p$. Write $p$ in the quotient.
2. Multiply the divisor ($p-3$) by the term just written in the quotient ($p$). $p(p-3) = p^2 - 3p$. Write this below the dividend.
3. Subtract the result from the dividend. $(p^2 - 8p) - (p^2 - 3p) = p^2 - 8p - p^2 + 3p = -5p$. Bring down the next term ($+15$) from the dividend. We now have $-5p + 15$.
4. Divide the first term of the new dividend ($-5p$) by the first term of the divisor ($p$). $-5p / p = -5$. Write $-5$ in the quotient.
5. Multiply the divisor ($p-3$) by the new term in the quotient ($-5$). $-5(p-3) = -5p + 15$. Write this below the current dividend.
6. Subtract the result. $(-5p + 15) - (-5p + 15) = -5p + 15 + 5p - 15 = 0$. The remainder is 0.

The quotient is $p - 5$ and the remainder is $0$.

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Alternatively, we can factor the quadratic expression $p^2 - 8p + 15$. We look for two numbers that multiply to $15$ and add up to $-8$. These numbers are $-3$ and $-5$.

$p^2 - 8p + 15 = (p - 3)(p - 5)$

Now, perform the division:

$\frac{(p - 3)(p - 5)}{(p - 3)}$

Assuming $p - 3 \neq 0$, we can cancel the common factor $(p - 3)$ from the numerator and the denominator:

$\frac{\cancel{(p - 3)}(p - 5)}{\cancel{(p - 3)}} = p - 5$

Using cancellation marks as requested:

$\frac{\cancel{(p - 3)} (p - 5)}{\cancel{(p - 3)}} = p - 5$

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The result of dividing $(p^2 - 8p + 15)$ by $(p - 3)$ is $p - 5$.

The correct options are (C) and (D).

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A binomial is an algebraic expression that consists of exactly two terms.

Let's examine each option and determine the number of terms:

(A) $a+b+c$: This expression has three terms ($a$, $b$, and $c$). It is a trinomial.

(B) $xy$: This expression is a product of two variables, $x$ and $y$. It forms a single term. It is a monomial.

(C) $m^2 - n^2$: This expression can be written as $m^2 + (-n^2)$. It has two terms ($m^2$ and $-n^2$). It is a binomial.

(D) $p - q$: This expression can be written as $p + (-q)$. It has two terms ($p$ and $-q$). It is a binomial.

(E) $5$: This is a constant term. It has one term. It is a monomial (a special case of a polynomial with degree 0).

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The expressions that are binomials are $m^2 - n^2$ and $p - q$.

The correct option is (D).

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Let a monomial be $M$ (1 term) and a trinomial be $T$ (3 terms).

The product $M \times T$ is found by multiplying $M$ by each term of $T$ and adding the results.

Since the product of two monomials is a monomial, the result is the sum of three monomials.

When these monomials are added, like terms might combine, but the resulting polynomial will have a maximum of 3 terms.

If the multiplying monomial is 0, the result is 0 (a monomial, 1 term).

In other cases, the result is typically a trinomial (3 terms), but could potentially simplify to a binomial (2 terms) in specific scenarios.

A polynomial with at most 3 terms includes monomials, binomials, and trinomials.

The correct option is (C).

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To simplify $-(a-b)$, distribute the negative sign (which is like multiplying by $-1$) to each term inside the parentheses.

$-(a-b) = -1 \times (a - b)$

$= (-1 \times a) + (-1 \times -b)$

$= -a + b$

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The simplified expression is $-a+b$.

The correct option is (A).

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We are given the expressions $A = x^2 + 2x + 1$ and $B = x^2 - 2x + 1$.

We need to find the value of $A - B$.

$A - B = (x^2 + 2x + 1) - (x^2 - 2x + 1)$

To subtract the second polynomial, change the sign of each term inside the parentheses and then add:

$A - B = x^2 + 2x + 1 - x^2 + 2x - 1$

Group the like terms:

$A - B = (x^2 - x^2) + (2x + 2x) + (1 - 1)$

Combine the like terms:

$A - B = 0x^2 + 4x + 0$

$A - B = 4x$

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The result of $A - B$ is $4x$.

An algebraic expression is a combination of constants, variables, and algebraic operations (such as addition, subtraction, multiplication, and division).

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Example:

$3x^2 + 2y - 5$

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In the given example $3x^2 + 2y - 5$:

Variable(s): $x$ and $y$. Variables are symbols (usually letters) that represent unknown values or values that can change.

Constant(s): $3$, $2$, and $-5$. Constants are values that do not change. In the terms $3x^2$ and $2y$, $3$ and $2$ are coefficients of the variables, but they are also constant numerical values within the expression. $-5$ is a standalone constant term.

The terms of an algebraic expression are the parts separated by addition or subtraction signs. The given expression is $7x^2y - 5xy + 9y^2 - 11$.

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The terms of the expression $7x^2y - 5xy + 9y^2 - 11$ are:

$7x^2y$

$-5xy$

$9y^2$

$-11$

Like Terms:

Terms that have the same variables raised to the same powers are called like terms. Only the numerical coefficients may be different.

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Example of Like Terms:

$5x^2y$ and $-7x^2y$. Both terms have the variables $x$ and $y$, with $x$ raised to the power of $2$ and $y$ raised to the power of $1$.

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Unlike Terms:

Terms that have different variables or the same variables raised to different powers are called unlike terms. Unlike terms cannot be combined through addition or subtraction.

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Example of Unlike Terms:

$5x^2y$ and $5xy^2$. Both terms have the variables $x$ and $y$, but the powers are different ($x^2$ vs $x^1$, $y^1$ vs $y^2$).

Another example: $3a$ and $4b$. These terms have different variables ($a$ vs $b$).

To identify like terms, we look for terms that have the same variables raised to the same powers.

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Let's examine each term:

The terms with $a^2b$ are $4a^2b$, $3a^2b$, and $a^2b$. These are like terms.

The terms with $ab^2$ are $-5ab^2$ and $-10ab^2$. These are like terms.

The terms with $ab$ are $8ab$ and $2ab$. These are like terms.

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The like terms from the given list are:

Group 1: $4a^2b$, $3a^2b$, $a^2b$

Group 2: $-5ab^2$, $-10ab^2$

Group 3: $8ab$, $2ab$

The given term is $-8x^2y^3$. A coefficient of a variable (or a product of variables) in a term is the remaining factor(s) after removing the variable(s) in question.

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In the term $-8x^2y^3$, to find the coefficient of $x^2$, we remove $x^2$. The remaining factors are $-8$ and $y^3$.

The coefficient of $x^2$ in $-8x^2y^3$ is $-8y^3$.

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In the same term $-8x^2y^3$, to find the coefficient of $y^3$, we remove $y^3$. The remaining factors are $-8$ and $x^2$.

The coefficient of $y^3$ in $-8x^2y^3$ is $-8x^2$.

Algebraic expressions are classified based on the number of terms they contain:

A monomial has exactly one term.

A binomial has exactly two terms.

A trinomial has exactly three terms.

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(a) The expression is $5x + 2y$. This expression has two terms: $5x$ and $2y$.

Therefore, $5x + 2y$ is a binomial.

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(b) The expression is $-7a^2b$. This expression has only one term: $-7a^2b$.

Therefore, $-7a^2b$ is a monomial.

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(c) The expression is $p^2 - 3p + 7$. This expression has three terms: $p^2$, $-3p$, and $7$.

Therefore, $p^2 - 3p + 7$ is a trinomial.

To add the expressions $(7m - 3n)$ and $(5m + 8n)$, we combine the like terms.

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$(7m - 3n) + (5m + 8n)$

Group the terms with $m$ and the terms with $n$ together:

$(7m + 5m) + (-3n + 8n)$

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Combine the coefficients of the like terms:

$(7 + 5)m + (-3 + 8)n$

$12m + 5n$

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The sum of the expressions is $12m + 5n$.

To subtract $(2x^2 - 5y)$ from $(6x^2 + 2y)$, we write the expression as:

$(6x^2 + 2y) - (2x^2 - 5y)$

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When subtracting an expression, we change the sign of each term being subtracted and then add the resulting terms.

$(6x^2 + 2y) + (-1) \times (2x^2 - 5y)$

$(6x^2 + 2y) + (-2x^2 + 5y)$

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Now, group the like terms:

$(6x^2 - 2x^2) + (2y + 5y)$

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Combine the coefficients of the like terms:

$(6 - 2)x^2 + (2 + 5)y$

$4x^2 + 7y$

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The result of the subtraction is $4x^2 + 7y$.

We are asked to simplify the expression: $(4a + 5b - 3c) + (2a - b + 4c) - (a + 2b - c)$.

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First, remove the parentheses. Remember to change the sign of each term inside the parentheses that are preceded by a subtraction sign:

$4a + 5b - 3c + 2a - b + 4c - a - 2b + c$

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Next, group the like terms together:

$(4a + 2a - a) + (5b - b - 2b) + (-3c + 4c + c)$

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Combine the coefficients of the like terms:

$(4 + 2 - 1)a + (5 - 1 - 2)b + (-3 + 4 + 1)c$

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Perform the arithmetic on the coefficients:

$(6 - 1)a + (4 - 2)b + (1 + 1)c$

$5a + 2b + 2c$

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The simplified expression is $5a + 2b + 2c$.

We need to evaluate the expression $3x^2 - 5x + 7$ when $x = 2$.

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Substitute the value $x = 2$ into the expression:

$3(2)^2 - 5(2) + 7$

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First, calculate the power of $x$:

$3(4) - 5(2) + 7$

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Next, perform the multiplications:

$12 - 10 + 7$

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Finally, perform the additions and subtractions from left to right:

$2 + 7$

$9$

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The value of the expression $3x^2 - 5x + 7$ when $x = 2$ is $9$.

We need to find the value of the expression $2a^2 - 3ab + b^2$ when $a = 1$ and $b = -1$.

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Substitute the values $a = 1$ and $b = -1$ into the expression:

$2(1)^2 - 3(1)(-1) + (-1)^2$

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Calculate the powers:

$(1)^2 = 1$

$(-1)^2 = (-1) \times (-1) = 1$

So the expression becomes:

$2(1) - 3(1)(-1) + 1$

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Perform the multiplications:

$2 \times 1 = 2$

$3 \times 1 \times (-1) = 3 \times (-1) = -3$

So the expression becomes:

$2 - (-3) + 1$

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Simplify the subtraction of a negative number:

$2 + 3 + 1$

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Perform the additions:

$5 + 1$

$6$

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The value of the expression $2a^2 - 3ab + b^2$ when $a = 1$ and $b = -1$ is $6$.

The phrase "the sum of squares of $x$ and $y$" can be broken down:

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"squares of $x$" means squaring the variable $x$, which is written as $x^2$.

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"squares of $y$" means squaring the variable $y$, which is written as $y^2$.

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"the sum of" means adding the two quantities that follow.

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So, "the sum of squares of $x$ and $y$" means adding $x^2$ and $y^2$.

The algebraic expression is $x^2 + y^2$.

The phrase "the difference of $p$ and $q$" means subtracting $q$ from $p$. This can be written as $p - q$.

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"Four times" this difference means multiplying the difference by 4.

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To multiply the entire difference by 4, we enclose the difference in parentheses:

$4 \times (p - q)$

Or simply:

$4(p - q)$

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The algebraic expression for "four times the difference of $p$ and $q$" is $4(p - q)$.

To multiply the monomial $5x^2$ by the binomial $(3x - 4y)$, we use the distributive property. This means we multiply the monomial by each term inside the binomial separately and then add the results.

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$5x^2 \times (3x - 4y)$

$= (5x^2 \times 3x) + (5x^2 \times -4y)$

---

Now, perform the multiplications for each term:

$5x^2 \times 3x = (5 \times 3) \times (x^2 \times x) = 15x^{2+1} = 15x^3$

$5x^2 \times -4y = (5 \times -4) \times (x^2 \times y) = -20x^2y$

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Combine the results:

$15x^3 + (-20x^2y)$

$15x^3 - 20x^2y$

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The product of $5x^2$ and $(3x - 4y)$ is $15x^3 - 20x^2y$.

To find the product of $(a + 7)$ and $(a - 3)$, we can use the distributive property, multiplying each term in the first binomial by each term in the second binomial.

---

$(a + 7)(a - 3)$

Multiply the first term of the first binomial ($a$) by each term of the second binomial:

$a \times a = a^2$

$a \times -3 = -3a$

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Multiply the second term of the first binomial ($+7$) by each term of the second binomial:

$7 \times a = 7a$

$7 \times -3 = -21$

---

Now, combine all the resulting terms:

$a^2 - 3a + 7a - 21$

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Combine the like terms ($-3a$ and $7a$):

$-3a + 7a = (-3 + 7)a = 4a$

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Substitute this back into the expression:

$a^2 + 4a - 21$

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The product of $(a + 7)$ and $(a - 3)$ is $a^2 + 4a - 21$.

We can expand the expression $(x + 5)^2$ using the algebraic identity for the square of a binomial:

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The identity is: $(a + b)^2 = a^2 + 2ab + b^2$

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In the given expression $(x + 5)^2$, we can identify $a = x$ and $b = 5$.

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Substitute these values into the identity:

$(x + 5)^2 = (x)^2 + 2(x)(5) + (5)^2$

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Perform the calculations for each term:

$(x)^2 = x^2$

$2(x)(5) = 10x$

$(5)^2 = 5 \times 5 = 25$

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Combine the terms to get the expanded form:

$x^2 + 10x + 25$

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Therefore, the expanded form of $(x + 5)^2$ is $x^2 + 10x + 25$.

We can expand the expression $(3y - 4)^2$ using the algebraic identity for the square of a binomial difference:

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The identity is: $(a - b)^2 = a^2 - 2ab + b^2$

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In the given expression $(3y - 4)^2$, we can identify $a = 3y$ and $b = 4$.

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Substitute these values into the identity:

$(3y - 4)^2 = (3y)^2 - 2(3y)(4) + (4)^2$

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Perform the calculations for each term:

$(3y)^2 = (3)^2 \times (y)^2 = 9y^2$

$2(3y)(4) = 2 \times 3 \times 4 \times y = 24y$

$(4)^2 = 4 \times 4 = 16$

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Combine the terms to get the expanded form:

$9y^2 - 24y + 16$

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Therefore, the expanded form of $(3y - 4)^2$ is $9y^2 - 24y + 16$.

We need to find the product of $(2p + 3q)$ and $(2p - 3q)$. This expression is in the form of the difference of squares identity.

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The algebraic identity for the difference of squares is: $(a + b)(a - b) = a^2 - b^2$

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In the given expression $(2p + 3q)(2p - 3q)$, we can identify $a = 2p$ and $b = 3q$.

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Substitute these values of $a$ and $b$ into the identity $a^2 - b^2$:

$(2p + 3q)(2p - 3q) = (2p)^2 - (3q)^2$

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Calculate the square of each term:

$(2p)^2 = (2)^2 \times (p)^2 = 4p^2$

$(3q)^2 = (3)^2 \times (q)^2 = 9q^2$

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Substitute these results back into the expression:

$4p^2 - 9q^2$

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The product of $(2p + 3q)$ and $(2p - 3q)$ is $4p^2 - 9q^2$.

To divide the binomial $(12x^3 + 8x^2)$ by the monomial $(4x)$, we can divide each term in the binomial by the monomial.

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Write the division as a fraction:

$\frac{12x^3 + 8x^2}{4x}$

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Separate the terms in the numerator and divide each by the denominator:

$\frac{12x^3}{4x} + \frac{8x^2}{4x}$

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Simplify each term:

For the first term, $\frac{12x^3}{4x}$:

Divide the coefficients: $\frac{12}{4} = 3$

Divide the variables using the rule $\frac{a^m}{a^n} = a^{m-n}$: $\frac{x^3}{x^1} = x^{3-1} = x^2$

So, $\frac{12x^3}{4x} = 3x^2$

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For the second term, $\frac{8x^2}{4x}$:

Divide the coefficients: $\frac{8}{4} = 2$

Divide the variables: $\frac{x^2}{x^1} = x^{2-1} = x^1 = x$

So, $\frac{8x^2}{4x} = 2x$

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Combine the simplified terms:

$3x^2 + 2x$

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The result of the division is $3x^2 + 2x$.

To divide the trinomial $(18a^4 - 27a^3 + 36a^2)$ by the monomial $(9a^2)$, we can divide each term in the trinomial by the monomial.

---

Write the division as a fraction:

$\frac{18a^4 - 27a^3 + 36a^2}{9a^2}$

---

Separate the terms in the numerator and divide each by the denominator:

$\frac{18a^4}{9a^2} - \frac{27a^3}{9a^2} + \frac{36a^2}{9a^2}$

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Simplify each term:

For the first term, $\frac{18a^4}{9a^2}$:

Divide the coefficients: $\frac{18}{9} = 2$

Divide the variables using the rule $\frac{a^m}{a^n} = a^{m-n}$: $\frac{a^4}{a^2} = a^{4-2} = a^2$

So, $\frac{18a^4}{9a^2} = 2a^2$

---

For the second term, $\frac{27a^3}{9a^2}$:

Divide the coefficients: $\frac{27}{9} = 3$

Divide the variables: $\frac{a^3}{a^2} = a^{3-2} = a^1 = a$

So, $\frac{27a^3}{9a^2} = 3a$

---

For the third term, $\frac{36a^2}{9a^2}$:

Divide the coefficients: $\frac{36}{9} = 4$

Divide the variables: $\frac{a^2}{a^2} = a^{2-2} = a^0 = 1$ (for $a \neq 0$)

So, $\frac{36a^2}{9a^2} = 4 \times 1 = 4$

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Combine the simplified terms:

$2a^2 - 3a + 4$

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The result of the division is $2a^2 - 3a + 4$.

We need to simplify the expression $2(x^2 + 3x) - 3(x^2 - x)$. We will use the distributive property to remove the parentheses.

---

Apply the distributive property to the first term, $2(x^2 + 3x)$:

$2 \times x^2 + 2 \times 3x = 2x^2 + 6x$

---

Apply the distributive property to the second term, $-3(x^2 - x)$. Remember to multiply $-3$ by each term inside the parentheses:

$-3 \times x^2 + (-3) \times (-x) = -3x^2 + 3x$

---

Now, combine the results from both terms:

$(2x^2 + 6x) + (-3x^2 + 3x)$

$2x^2 + 6x - 3x^2 + 3x$

---

Group the like terms (terms with $x^2$ and terms with $x$):

$(2x^2 - 3x^2) + (6x + 3x)$

---

Combine the coefficients of the like terms:

$(2 - 3)x^2 + (6 + 3)x$

$-1x^2 + 9x$

$-x^2 + 9x$

---

The simplified expression is $-x^2 + 9x$.

The given algebraic expression is $x^3 - 5x^2 + 7x - 11$.

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The terms of the expression are $x^3$, $-5x^2$, $7x$, and $-11$.

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The coefficient of $x^2$ is the numerical factor of the term containing $x^2$. In the term $-5x^2$, the coefficient of $x^2$ is $-5$.

The coefficient of $x^2$ is $-5$.

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The constant term in an algebraic expression is the term that does not contain any variables. In the given expression, the term without any variable is $-11$.

The constant term is $-11$.

We need to evaluate the expression $m^2 - 2mn + n^2$ when $m = 5$ and $n = 3$.

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Substitute the values $m = 5$ and $n = 3$ into the expression:

$(5)^2 - 2(5)(3) + (3)^2$

---

Calculate the powers:

$(5)^2 = 5 \times 5 = 25$

$(3)^2 = 3 \times 3 = 9$

So the expression becomes:

$25 - 2(5)(3) + 9$

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Perform the multiplication:

$2(5)(3) = 2 \times 5 \times 3 = 10 \times 3 = 30$

So the expression becomes:

$25 - 30 + 9$

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Perform the subtraction and addition from left to right:

$25 - 30 = -5$

$-5 + 9 = 4$

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The value of the expression $m^2 - 2mn + n^2$ when $m = 5$ and $n = 3$ is $4$.

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Alternate Method (using identity):

The expression $m^2 - 2mn + n^2$ is an algebraic identity, which is equal to $(m - n)^2$.

---

Substitute the values $m = 5$ and $n = 3$ into the identity $(m - n)^2$:

$(5 - 3)^2$

---

Perform the subtraction inside the parentheses:

$2^2$

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Calculate the square:

$2^2 = 2 \times 2 = 4$

---

Using the identity, the value of the expression is also $4$.

The cost of one notebook is given as $\textsf{₹}(2x + 5)$.

---

To find the cost of 8 such notebooks, we need to multiply the cost of one notebook by the number of notebooks.

---

Cost of 8 notebooks = $8 \times (\textsf{₹}(2x + 5))$

Cost of 8 notebooks = $\textsf{₹} \times 8 \times (2x + 5)$

---

Using the distributive property, we multiply 8 by each term inside the parentheses:

$8 \times (2x + 5) = (8 \times 2x) + (8 \times 5)$

$= 16x + 40$

---

So, the algebraic expression for the cost of 8 such notebooks is $\textsf{₹}(16x + 40)$.

The expression is $\textsf{₹}(16x + 40)$.

To find the product of $2ab$ and $(-3a^2c)$, we multiply the numerical coefficients and the variable parts separately.

---

The expression for the product is:

$(2ab) \times (-3a^2c)$

---

Multiply the numerical coefficients: $2 \times (-3) = -6$

---

Multiply the variable parts: $ab \times a^2c$

Combine the powers of the same variables using the rule $x^m \times x^n = x^{m+n}$:

$a^1 \times a^2 = a^{1+2} = a^3$

$b$ remains as $b$ (since there is no $b$ in the second term)

$c$ remains as $c$ (since there is no $c$ in the first term)

So, the product of the variable parts is $a^3bc$.

---

Combine the numerical coefficient and the variable part:

$-6 \times a^3bc = -6a^3bc$

---

The product of $2ab$ and $(-3a^2c)$ is $-6a^3bc$.

The degree of a polynomial is the highest degree among all its terms.

The degree of a term with more than one variable is the sum of the exponents of the variables in that term. For a term with a single variable, it is the exponent of the variable. The degree of a non-zero constant term is 0.

---

The given polynomial is $5x^3y^2 - 7x^4 + 2y^5 - 1$.

Let's find the degree of each term:

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1. Term: $5x^3y^2$

The variables are $x$ and $y$. The exponent of $x$ is 3 and the exponent of $y$ is 2.

Degree of this term = $3 + 2 = 5$

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2. Term: $-7x^4$

The variable is $x$. The exponent of $x$ is 4.

Degree of this term = $4$

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3. Term: $2y^5$

The variable is $y$. The exponent of $y$ is 5.

Degree of this term = $5$

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4. Term: $-1$

This is a constant term.

Degree of this term = $0$

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The degrees of the terms are 5, 4, 5, and 0.

The highest degree among these is 5.

---

Therefore, the degree of the polynomial $5x^3y^2 - 7x^4 + 2y^5 - 1$ is 5.

No, the expression $x^2 - \frac{1}{x}$ is not a polynomial.

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A polynomial is an algebraic expression where the exponents of the variables are always non-negative integers ($0, 1, 2, 3, ...$).

---

The given expression is $x^2 - \frac{1}{x}$.

We can rewrite the term $\frac{1}{x}$ using a negative exponent:

$\frac{1}{x} = x^{-1}$

---

So the expression can be written as $x^2 - x^{-1}$.

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In this expression, the exponent of the variable $x$ in the second term ($x^{-1}$) is $-1$, which is a negative integer.

Since the exponent is not a non-negative integer, the expression $x^2 - \frac{1}{x}$ does not satisfy the condition for being a polynomial.

Step 1: Find the sum of $(5x^2 - 3x + 1)$ and $(-2x^2 + 7x - 5)$.

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Sum = $(5x^2 - 3x + 1) + (-2x^2 + 7x - 5)$

Remove the parentheses:

Sum = $5x^2 - 3x + 1 - 2x^2 + 7x - 5$

---

Group the like terms:

Sum = $(5x^2 - 2x^2) + (-3x + 7x) + (1 - 5)$

---

Combine the coefficients of the like terms:

Sum = $(5 - 2)x^2 + (-3 + 7)x + (1 - 5)$

Sum = $3x^2 + 4x - 4$

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Step 2: Subtract $(x^2 - x - 4)$ from the sum $3x^2 + 4x - 4$.

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Difference = $(3x^2 + 4x - 4) - (x^2 - x - 4)$

To subtract, change the sign of each term in the second polynomial and add:

Difference = $3x^2 + 4x - 4 - x^2 + x + 4$

---

Group the like terms:

Difference = $(3x^2 - x^2) + (4x + x) + (-4 + 4)$

---

Combine the coefficients of the like terms:

Difference = $(3 - 1)x^2 + (4 + 1)x + (-4 + 4)$

Difference = $2x^2 + 5x + 0$

Difference = $2x^2 + 5x$

---

The result is $2x^2 + 5x$.

We need to simplify the expression: $3(2a^2 - 5a + 1) - 2(4a^2 + 3a - 5) - (a^2 - a + 2)$.

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First, distribute the coefficients $3$, $-2$, and $-1$ (for the last parenthesis) to the terms inside their respective parentheses:

$3 \times (2a^2 - 5a + 1) = 3 \times 2a^2 + 3 \times (-5a) + 3 \times 1 = 6a^2 - 15a + 3$

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$-2 \times (4a^2 + 3a - 5) = -2 \times 4a^2 + (-2) \times 3a + (-2) \times (-5) = -8a^2 - 6a + 10$

---

$-(a^2 - a + 2) = -1 \times (a^2 - a + 2) = -1 \times a^2 + (-1) \times (-a) + (-1) \times 2 = -a^2 + a - 2$

---

Now, rewrite the expression with the distributed terms:

$(6a^2 - 15a + 3) + (-8a^2 - 6a + 10) + (-a^2 + a - 2)$

$6a^2 - 15a + 3 - 8a^2 - 6a + 10 - a^2 + a - 2$

---

Group the like terms together:

$(6a^2 - 8a^2 - a^2) + (-15a - 6a + a) + (3 + 10 - 2)$

---

Combine the coefficients of the like terms:

$(6 - 8 - 1)a^2 + (-15 - 6 + 1)a + (3 + 10 - 2)$

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Perform the arithmetic on the coefficients:

$(6 - 9)a^2 + (-21 + 1)a + (13 - 2)$

$-3a^2 + (-20)a + 11$

$-3a^2 - 20a + 11$

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The simplified expression is $-3a^2 - 20a + 11$.

We need to find the product of the binomial $(3x + 2y)$ and the trinomial $(9x^2 - 6xy + 4y^2)$. We will use the distributive property to multiply each term in the first expression by each term in the second expression.

---

$(3x + 2y)(9x^2 - 6xy + 4y^2)$

---

Multiply the first term of the binomial ($3x$) by each term of the trinomial:

$3x \times (9x^2 - 6xy + 4y^2)$

$= (3x \times 9x^2) + (3x \times -6xy) + (3x \times 4y^2)$

$= 27x^3 - 18x^2y + 12xy^2$

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Multiply the second term of the binomial ($+2y$) by each term of the trinomial:

$+2y \times (9x^2 - 6xy + 4y^2)$

$= (2y \times 9x^2) + (2y \times -6xy) + (2y \times 4y^2)$

$= 18x^2y - 12xy^2 + 8y^3$

---

Now, combine all the resulting terms from the multiplications:

$27x^3 - 18x^2y + 12xy^2 + 18x^2y - 12xy^2 + 8y^3$

---

Group the like terms together:

$27x^3 + (-18x^2y + 18x^2y) + (12xy^2 - 12xy^2) + 8y^3$

---

Combine the like terms:

$-18x^2y + 18x^2y = 0$

$12xy^2 - 12xy^2 = 0$

---

The expression simplifies to:

$27x^3 + 0 + 0 + 8y^3$

$27x^3 + 8y^3$

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Alternatively, one can recognize that the product is in the form of the sum of cubes identity: $(a+b)(a^2 - ab + b^2) = a^3 + b^3$.

Here, $a = 3x$ and $b = 2y$.

The product is $(3x)^3 + (2y)^3 = 27x^3 + 8y^3$.

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The simplified product is $27x^3 + 8y^3$.

We need to evaluate the expression $x^3 + 3x^2 - 5x + 8$ when $x = -3$.

---

Substitute $x = -3$ into the expression:

$(-3)^3 + 3(-3)^2 - 5(-3) + 8$

---

Calculate the powers of $-3$:

$(-3)^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27$

$(-3)^2 = (-3) \times (-3) = 9$

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Substitute these values back into the expression:

$-27 + 3(9) - 5(-3) + 8$

---

Perform the multiplications:

$3(9) = 27$

$-5(-3) = 15$

---

Substitute these values back into the expression:

$-27 + 27 + 15 + 8$

---

Perform the addition and subtraction from left to right:

$-27 + 27 = 0$

$0 + 15 = 15$

$15 + 8 = 23$

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The value of the expression $x^3 + 3x^2 - 5x + 8$ for $x = -3$ is $23$.

We need to evaluate $(107)^2$ using the identity $(a+b)^2 = a^2 + 2ab + b^2$.

---

We are instructed to write $107$ as $(100+7)$.

So, we have $107 = 100 + 7$.

---

Comparing $(100 + 7)^2$ with $(a+b)^2$, we can identify $a = 100$ and $b = 7$.

---

The identity is: $(a+b)^2 = a^2 + 2ab + b^2$.

---

Substituting $a = 100$ and $b = 7$ into the identity, we get:

$(107)^2 = (100 + 7)^2$

$(100 + 7)^2 = (100)^2 + 2(100)(7) + (7)^2$

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Now, we calculate the value of each term:

$a^2 = (100)^2 = 100 \times 100 = 10000$

$2ab = 2 \times 100 \times 7 = 200 \times 7 = 1400$

$b^2 = (7)^2 = 7 \times 7 = 49$

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Adding these values together to find the result of $(107)^2$:

$(107)^2 = 10000 + 1400 + 49$

$(107)^2 = 11400 + 49$

$(107)^2 = 11449$

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Thus, using the identity, we find that $(107)^2 = 11449$.

We need to evaluate $(96)^2$ using the identity $(a-b)^2 = a^2 - 2ab + b^2$.

---

We are instructed to write $96$ as $(100-4)$.

So, we have $96 = 100 - 4$.

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Comparing $(100 - 4)^2$ with $(a-b)^2$, we can identify $a = 100$ and $b = 4$.

---

The identity is: $(a-b)^2 = a^2 - 2ab + b^2$.

---

Substituting $a = 100$ and $b = 4$ into the identity, we get:

$(96)^2 = (100 - 4)^2$

$(100 - 4)^2 = (100)^2 - 2(100)(4) + (4)^2$

---

Now, we calculate the value of each term:

$a^2 = (100)^2 = 100 \times 100 = 10000$

$2ab = 2 \times 100 \times 4 = 200 \times 4 = 800$

$b^2 = (4)^2 = 4 \times 4 = 16$

---

Performing the subtraction and addition as per the identity:

$(96)^2 = 10000 - 800 + 16$

$(96)^2 = 9200 + 16$

$(96)^2 = 9216$

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Thus, using the identity, we find that $(96)^2 = 9216$.

We need to evaluate $54 \times 46$ using the identity $(a+b)(a-b) = a^2 - b^2$.

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We are instructed to write $54$ as $(50+4)$ and $46$ as $(50-4)$.

So, we have $54 = 50 + 4$ and $46 = 50 - 4$.

---

The expression $54 \times 46$ can be written as $(50+4) \times (50-4)$.

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Comparing $(50+4)(50-4)$ with the identity $(a+b)(a-b)$, we can identify $a = 50$ and $b = 4$.

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The identity is: $(a+b)(a-b) = a^2 - b^2$.

---

Substituting $a = 50$ and $b = 4$ into the identity, we get:

$54 \times 46 = (50+4)(50-4)$

$(50+4)(50-4) = (50)^2 - (4)^2$

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Now, we calculate the value of $a^2$ and $b^2$:

$a^2 = (50)^2 = 50 \times 50 = 2500$

$b^2 = (4)^2 = 4 \times 4 = 16$

---

Performing the subtraction as per the identity:

$54 \times 46 = 2500 - 16$

$2500 - 16 = 2484$

---

Thus, using the identity, we find that $54 \times 46 = 2484$.

We need to verify the identity $(a+b)^2 = a^2 + 2ab + b^2$ for $a = 3$ and $b = -2$.

---

Given:

$a = 3$

$b = -2$

---

Identity to verify:

$(a+b)^2 = a^2 + 2ab + b^2$

---

Calculate LHS $(a+b)^2$:

Substitute $a=3$ and $b=-2$ into the LHS:

LHS $= (3 + (-2))^2$

LHS $= (3 - 2)^2$

LHS $= (1)^2$

LHS $= 1$

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Calculate RHS $a^2 + 2ab + b^2$:

Substitute $a=3$ and $b=-2$ into the RHS:

RHS $= (3)^2 + 2(3)(-2) + (-2)^2$

RHS $= 9 + 2(-6) + 4$

RHS $= 9 - 12 + 4$

RHS $= -3 + 4$

RHS $= 1$

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Verification:

We found that LHS $= 1$ and RHS $= 1$.

Since LHS $=$ RHS, the identity $(a+b)^2 = a^2 + 2ab + b^2$ is verified for $a = 3$ and $b = -2$.

We are given the dimensions of a rectangle in terms of $x$.

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Given:

Length of the rectangle $= (3x+5)$ meters

Breadth of the rectangle $= (2x-1)$ meters

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To Find:

1. Expression for the area of the rectangle.

2. Area of the rectangle when $x=4$ meters.

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Solution:

The formula for the area of a rectangle is:

Area $=$ Length $\times$ Breadth

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Substitute the given expressions for length and breadth into the formula:

Area $= (3x+5) \times (2x-1)$

Expand the expression using the distributive property:

Area $= 3x(2x-1) + 5(2x-1)$

Area $= (3x \times 2x) + (3x \times -1) + (5 \times 2x) + (5 \times -1)$

Area $= 6x^2 - 3x + 10x - 5$

Combine the like terms ($-3x$ and $10x$):

Area $= 6x^2 + (10x - 3x) - 5$

Area $= 6x^2 + 7x - 5$

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So, the expression for the area of the rectangle is $(6x^2 + 7x - 5)$ square meters.

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Now, we need to find the area when $x = 4$ meters.

Substitute $x=4$ into the area expression:

Area $= 6(4)^2 + 7(4) - 5$

Calculate the terms:

Area $= 6(16) + 28 - 5$

Area $= 96 + 28 - 5$

Perform the addition and subtraction:

Area $= 124 - 5$

Area $= 119$

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Therefore, the area of the rectangle when $x=4$ meters is $119$ square meters.

We need to divide the polynomial $(20x^3y - 30x^2y^2 + 40xy^3)$ by the monomial $(5xy)$.

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This can be written as:

$\frac{20x^3y - 30x^2y^2 + 40xy^3}{5xy}$

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To divide a polynomial by a monomial, we divide each term of the polynomial by the monomial.

So, we divide each term separately:

$\frac{20x^3y}{5xy} - \frac{30x^2y^2}{5xy} + \frac{40xy^3}{5xy}$

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Division of the first term:

Divide $20x^3y$ by $5xy$:

$\frac{20x^3y}{5xy} = \frac{20}{5} \times \frac{x^3}{x} \times \frac{y}{y}$

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$ and $\frac{y}{y} = y^{1-1} = y^0 = 1$ (for $y \neq 0$):

$\frac{20}{5} = 4$

$\frac{x^3}{x} = x^{3-1} = x^2$

$\frac{y}{y} = 1$

So, $\frac{20x^3y}{5xy} = 4 \times x^2 \times 1 = 4x^2$.

Alternatively, showing cancellation:

$\frac{\cancel{20}^{4} x^{\cancel{3}^{2}} \cancel{y}}{\cancel{5}_{1} \cancel{x}_{1} \cancel{y}_{1}} = 4x^2$

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Division of the second term:

Divide $-30x^2y^2$ by $5xy$:

$\frac{-30x^2y^2}{5xy} = \frac{-30}{5} \times \frac{x^2}{x} \times \frac{y^2}{y}$

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{-30}{5} = -6$

$\frac{x^2}{x} = x^{2-1} = x^1 = x$

$\frac{y^2}{y} = y^{2-1} = y^1 = y$

So, $\frac{-30x^2y^2}{5xy} = -6 \times x \times y = -6xy$.

Alternatively, showing cancellation:

$\frac{\cancel{-30}^{-6} x^{\cancel{2}^{1}} y^{\cancel{2}^{1}}}{\cancel{5}_{1} \cancel{x}_{1} \cancel{y}_{1}} = -6xy$

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Division of the third term:

Divide $40xy^3$ by $5xy$:

$\frac{40xy^3}{5xy} = \frac{40}{5} \times \frac{x}{x} \times \frac{y^3}{y}$

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$ and $\frac{x}{x} = x^{1-1} = x^0 = 1$ (for $x \neq 0$):

$\frac{40}{5} = 8$

$\frac{x}{x} = 1$

$\frac{y^3}{y} = y^{3-1} = y^2$

So, $\frac{40xy^3}{5xy} = 8 \times 1 \times y^2 = 8y^2$.

Alternatively, showing cancellation:

$\frac{\cancel{40}^{8} \cancel{x}_{1} y^{\cancel{3}^{2}}}{\cancel{5}_{1} \cancel{x}_{1} \cancel{y}_{1}} = 8y^2$

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Combining the results:

The result of the division is the sum of the results of dividing each term:

$\frac{20x^3y - 30x^2y^2 + 40xy^3}{5xy} = \frac{20x^3y}{5xy} - \frac{30x^2y^2}{5xy} + \frac{40xy^3}{5xy}$

$= 4x^2 - 6xy + 8y^2$

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The result of the division is $4x^2 - 6xy + 8y^2$.

We need to simplify the given expression: $(x + y)(x - y) + (y + z)(y - z) + (z + x)(z - x)$.

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We can use the identity $(a+b)(a-b) = a^2 - b^2$ to simplify each term of the expression.

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Consider the first term: $(x + y)(x - y)$.

Using the identity with $a=x$ and $b=y$, we get:

$(x + y)(x - y) = x^2 - y^2$

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Consider the second term: $(y + z)(y - z)$.

Using the identity with $a=y$ and $b=z$, we get:

$(y + z)(y - z) = y^2 - z^2$

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Consider the third term: $(z + x)(z - x)$.

Using the identity with $a=z$ and $b=x$, we get:

$(z + x)(z - x) = z^2 - x^2$

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Now, substitute these simplified terms back into the original expression:

$(x + y)(x - y) + (y + z)(y - z) + (z + x)(z - x) = (x^2 - y^2) + (y^2 - z^2) + (z^2 - x^2)$

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Remove the parentheses and group like terms:

$= x^2 - y^2 + y^2 - z^2 + z^2 - x^2$

$= (x^2 - x^2) + (-y^2 + y^2) + (-z^2 + z^2)$

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Combine the like terms:

$= 0 + 0 + 0$

$= 0$

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Thus, the simplified expression is $0$.

We need to first find the sum of the two expressions $(a^2 + ab + b^2)$ and $(a^2 - ab + b^2)$, and then subtract $(a^2 - b^2)$ from this sum.

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Step 1: Find the sum of $(a^2 + ab + b^2)$ and $(a^2 - ab + b^2)$.

Sum $= (a^2 + ab + b^2) + (a^2 - ab + b^2)$

Combine like terms:

Sum $= (a^2 + a^2) + (ab - ab) + (b^2 + b^2)$

Sum $= 2a^2 + 0 + 2b^2$

Sum $= 2a^2 + 2b^2$

The sum of the first two expressions is $2a^2 + 2b^2$.

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Step 2: Subtract $(a^2 - b^2)$ from the sum.

We need to calculate $(2a^2 + 2b^2) - (a^2 - b^2)$.

$(2a^2 + 2b^2) - (a^2 - b^2) = 2a^2 + 2b^2 - a^2 + b^2$

Combine like terms:

$= (2a^2 - a^2) + (2b^2 + b^2)$

$= a^2 + 3b^2$

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The result of the subtraction is $a^2 + 3b^2$.

Thus, $(a^2 + ab + b^2) + (a^2 - ab + b^2) - (a^2 - b^2) = \mathbf{a^2 + 3b^2}$.