Operations on Polynomials Practice Questions — ACT

Question 1Question →

A rectangular region has a width of $2x - 3$ meters and a length of $3x + 1$ meters. A square piece with a side length of $x - 2$ meters is removed from the region. The area, in square meters, of the remaining region can be expressed in the standard polynomial form $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: -3

Answer

The coefficient of the linear term,

B

, is

-3

.

Subtracting the area of the square,

x^2 - 4x + 4

, from the area of the rectangle,

6x^2 - 7x - 3

, yields

5x^2 - 3x - 7

. Thus, the coefficient

B

of the

x

term is

-3

.

Step-by-Step Solution

1

Calculate the area of the rectangle.

Area =

6x^2 - 7x - 3

The area of a rectangle is found by multiplying its length and width:

(2x - 3)(3x + 1) = 6x^2 + 2x - 9x - 3 = 6x^2 - 7x - 3

.

2

Calculate the area of the square.

Area =

x^2 - 4x + 4

The area of a square is the square of its side length:

(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4

.

3

Subtract the square's area from the rectangle's area.

Remaining Area =

5x^2 - 3x - 7

Subtracting the area of the removed square from the total area requires distributing the negative sign to each term of the square's polynomial:

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

.

4

Identify the coefficient

B

.

B = -3

In the standard quadratic form

Ax^2 + Bx + C

, the coefficient of the linear term

x

is

B

, which corresponds to

-3

in the polynomial

5x^2 - 3x - 7

.

Key Concept

Operations on Polynomials (multiplication, squaring binomials, and subtraction with negative sign distribution)

Question 2Question →

Which of the following expressions is equivalent to the expression below?

(3x^3 - 2y^2)^2 - 2(x^3 - 4y^2)(4x^3 - 5y^2)

x^9 + 30x^3y^2 - 36y^4

x^6 - 54x^3y^2 + 44y^4

x^6 + 30x^3y^2 - 36y^4

x^6 + 42x^3y^2 - 36y^4

-7x^6 + 72x^3y^2 - 76y^4

Show answer & explanation

Answer: $x^6 + 30x^3y^2 - 36y^4$

Answer

The expression

x^6 + 30x^3y^2 - 36y^4

.

To find the correct expression, we first square the binomial to obtain

9x^6 - 12x^3y^2 + 4y^4

. Next, we multiply the two binomials

(x^3 - 4y^2)(4x^3 - 5y^2)

to get

4x^6 - 21x^3y^2 + 20y^4

, and distribute the factor of 2 to obtain

8x^6 - 42x^3y^2 + 40y^4

. Finally, we subtract this from the squared binomial by distributing the negative sign to all terms:

9x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4

. Combining like terms yields

x^6 + 30x^3y^2 - 36y^4

.

Step-by-Step Solution

1

Expand the squared binomial

(3x^3 - 2y^2)^2

.

9x^6 - 12x^3y^2 + 4y^4

Use the binomial squaring formula

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

, where

a = 3x^3

and

b = 2y^2

.

2

Multiply the two binomials

(x^3 - 4y^2)(4x^3 - 5y^2)

.

4x^6 - 21x^3y^2 + 20y^4

Apply the FOIL method to expand the product:

4x^6 - 5x^3y^2 - 16x^3y^2 + 20y^4

, and combine the like middle terms.

3

Multiply the resulting trinomial from Step 2 by the constant factor of 2.

8x^6 - 42x^3y^2 + 40y^4

Distribute the factor of 2 to each term of the simplified trinomial expression.

4

Subtract the expression in Step 3 from the expression in Step 1.

x^6 + 30x^3y^2 - 36y^4

Distribute the negative sign to all terms of the subtracted polynomial:

9x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4

, and combine the remaining like terms.

Key Concept

Operations on Polynomials

Question 3Question →

When the product of the polynomials $(x^3 - 3x^2 + 2x - 4)$ and $(ax^2 + bx + c)$ is subtracted from $2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16$ , the resulting polynomial is equal to $0$ for all real values of $x$ . What is the value of $4a + 2b - c$ ?

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By equating the corresponding coefficients of the product

(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c)

to the polynomial

2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16

, we find

a = 2

from the

x^5

terms,

c = 4

from the constant terms, and

b = 1

from the

x^4

terms. Substituting these values into

4a + 2b - c

yields

4(2) + 2(1) - 4 = 6

.

Step-by-Step Solution

1

Relate the product of the polynomials to the given polynomial expression.

(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16

Since subtracting the product from the given polynomial results in a polynomial that is always

0

, the product must be identically equal to that polynomial.

2

Equate the leading coefficients to find the value of

a

.

a = 2

The highest-degree term of the product is

x^3 \cdot ax^2 = ax^5

, which must equal the highest-degree term on the right side,

2x^5

.

3

Equate the constant terms to find the value of

c

.

-4c = -16 \implies c = 4

The constant term of the product is

-4 \cdot c = -4c

, which must equal the constant term on the right side,

-16

.

4

Equate the coefficients of the

x^4

terms to find the value of

b

.

b - 3a = -5 \implies b - 6 = -5 \implies b = 1

The

x^4

term in the expanded product comes from

(x^3)(bx) + (-3x^2)(ax^2) = (b - 3a)x^4

, which must equal the

x^4

term on the right side,

-5x^4

.

5

Compute the value of the requested expression

4a + 2b - c

.

4(2) + 2(1) - 4 = 6

Substitute the determined values

a = 2

,

b = 1

, and

c = 4

into the expression.

Key Concept

Operations on polynomials, specifically multiplication, subtraction, and equating corresponding coefficients.

Alternative Method

Alternatively, evaluate the polynomial equation at

x = 2

. Substituting

x = 2

into

(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16

gives

(8 - 12 + 4 - 4)(4a + 2b + c) = 64 - 80 + 40 - 72 + 8 - 16

, which simplifies to

-4(4a + 2b + c) = -56

. Dividing both sides by

-4

yields

4a + 2b + c = 14

. Since equating the constant terms gives

-4c = -16 \implies c = 4

, we can substitute

c = 4

into

4a + 2b + c = 14

to get

4a + 2b + 4 = 14 \implies 4a + 2b = 10

. Subtracting

c=4

from both sides gives the desired expression value:

4a + 2b - c = 10 - 4 = 6

.

Estimated Time:2m 30s

Question 4Question →

A large rectangle has a length of $4x + 3$ inches and a width of $3x - 2$ inches. A smaller rectangular region is cut out from the center. The cut-out region has a length of $2x - 1$ inches and a width of $x - 5$ inches. The area of the remaining region can be written as the polynomial $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12, which is the coefficient of the linear term in the simplified remaining area polynomial.

The remaining area is found by subtracting the area of the smaller rectangle from the area of the larger rectangle. The area of the larger rectangle is

(4x + 3)(3x - 2) = 12x^2 + x - 6

. The area of the smaller rectangle is

(2x - 1)(x - 5) = 2x^2 - 11x + 5

. Subtracting the two yields

(12x^2 + x - 6) - (2x^2 - 11x + 5) = 10x^2 + 12x - 11

. Matching this to

Ax^2 + Bx + C

, the coefficient

B

of the linear term is

12

.

Step-by-Step Solution

1

Find the area of the outer rectangle.

Area =

12x^2 + x - 6

Multiply the length

(4x + 3)

and the width

(3x - 2)

using the distributive property.

2

Find the area of the inner rectangle.

Area =

2x^2 - 11x + 5

Multiply the length

(2x - 1)

and the width

(x - 5)

using the distributive property.

3

Subtract the inner area from the outer area to find the remaining area.

Remaining Area =

10x^2 + 12x - 11

Subtract the polynomial

(2x^2 - 11x + 5)

from

(12x^2 + x - 6)

by distributing the negative sign and combining like terms.

4

Identify the coefficient

B

of the

x

term.

B = 12

Match the simplified polynomial

10x^2 + 12x - 11

with the form

Ax^2 + Bx + C

.

Key Concept

Operations on Polynomials (multiplication and subtraction of polynomials)

Question 5Question →

A square has a side length of $x^3 - 3$ inches. Which of the following expressions represents the area, in square inches, of the square?

x^6 - 9

x^9 - 6x^3 + 9

x^6 - 6x^3 + 9

x^6 + 9

x^9 - 9

Show answer & explanation

Answer: $x^6 - 6x^3 + 9$

Answer

The expression

x^6 - 6x^3 + 9

To find the area of a square, square its side length:

(x^3 - 3)^2

. Expanding this gives

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

.

Step-by-Step Solution

1

Write the formula for the area of a square.

A = s^2

, where

s

is the side length.

The area of a square is the side length squared.

2

Substitute the given side length

s = x^3 - 3

into the area formula.

A = (x^3 - 3)^2

We replace

s

with the algebraic expression for the side length.

3

Expand the squared binomial

(x^3 - 3)^2

using the identity

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

.

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

Expanding the binomial yields the correct simplified polynomial in standard descending order.

Key Concept

Expanding a squared binomial and applying exponent rules for powers of powers.

Estimated Time:1m 0s

Question 6Question →

When the expression $(5x^2 - 8x + 2) - (2x^2 - 4x - 9)$ is written in the standard form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of $b$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

, which is the coefficient of the

x

term after simplifying the expression.

To subtract polynomials, we distribute the negative sign to all terms of the polynomial being subtracted and then combine like terms. Simplifying

(5x^2 - 8x + 2) - (2x^2 - 4x - 9)

gives

5x^2 - 8x + 2 - 2x^2 + 4x + 9 = 3x^2 - 4x + 11

. Comparing this to

ax^2 + bx + c

shows that

b

, the coefficient of the

x

term, is

-4

.

Step-by-Step Solution

1

Distribute the subtraction sign to all terms inside the second set of parentheses.

-2x^2 + 4x + 9

Subtracting a polynomial is equivalent to adding its opposite, which means changing the sign of every term in that polynomial.

2

Group and combine like terms from both polynomials.

3x^2 - 4x + 11

Like terms (terms with the same variable raised to the same power) can be combined by adding or subtracting their coefficients:

(5x^2 - 2x^2) = 3x^2

,

(-8x + 4x) = -4x

, and

(2 + 9) = 11

.

3

Compare the simplified expression to the standard form

ax^2 + bx + c

to identify the value of

b

.

b = -4

In the standard form

ax^2 + bx + c

, the coefficient of the

x

term is represented by

b

. In the simplified expression

3x^2 - 4x + 11

, the coefficient of

x

is

-4

.

Key Concept

Polynomial Subtraction and Combining Like Terms

Question 7Question →

A square metal sheet has a side length of $x + 4$ inches. A rectangular region with a width of $x$ inches and a length of $x + 2$ inches is removed from the sheet. Which of the following expressions represents the area, in square inches, of the remaining portion of the sheet?

-2x + 16

8x + 14

10x + 16

6x + 16

6x + 8

Show answer & explanation

Answer: 6x + 16

Answer

The expression that represents the remaining area is

6x + 16

.

To find the remaining area, subtract the area of the removed rectangle from the area of the original square sheet. The area of the square is

(x + 4)^2 = x^2 + 8x + 16

and the area of the rectangle is

x(x + 2) = x^2 + 2x

. Subtracting these yields

(x^2 + 8x + 16) - (x^2 + 2x) = x^2 + 8x + 16 - x^2 - 2x = 6x + 16

.

Step-by-Step Solution

1

Find the area of the original square metal sheet.

Area of the square is

(x + 4)^2 = x^2 + 8x + 16

square inches.

The area of a square is calculated by squaring its side length.

2

Find the area of the removed rectangular region.

Area of the rectangle is

x(x + 2) = x^2 + 2x

square inches.

The area of a rectangle is calculated by multiplying its width by its length.

3

Subtract the area of the rectangular region from the area of the square sheet.

(x^2 + 8x + 16) - (x^2 + 2x) = x^2 + 8x + 16 - x^2 - 2x = 6x + 16

square inches.

Subtracting the removed area from the total area gives the remaining area, distributing the negative sign to all terms inside the parentheses.

Key Concept

Operations on Polynomials

Estimated Time:45s

Question 8Question →

A square garden plot has a side length of $3s - 2$ yards. A walkway of uniform width $s + 3$ yards is built around the entire garden. Which of the following expressions represents the area, in square yards, of the walkway?

7s^2 + 20s - 3

16s^2 + 28s + 20

16s^2 + 52s + 12

16s^2 + 12

4s^2 + 52s + 12

Show answer & explanation

Answer: 16s^2 + 52s + 12

Answer

16s^2 + 52s + 12

The correct answer is found by first calculating the outer side length, which is the inner side length plus twice the walkway width:

(3s - 2) + 2(s + 3) = 5s + 4

. Squaring this yields the outer area of

25s^2 + 40s + 16

. The inner area is

(3s - 2)^2 = 9s^2 - 12s + 4

. Subtracting the inner area from the outer area and distributing the negative sign properly gives

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 16s^2 + 52s + 12

.

Step-by-Step Solution

1

Determine the outer side length of the square including the walkway.

The outer side length is

(3s - 2) + 2(s + 3) = 3s - 2 + 2s + 6 = 5s + 4

yards.

Since the walkway surrounds the garden on all sides, the width of the walkway must be added to both ends of the garden's side length.

2

Calculate the area of the outer square and the inner square garden by squaring their respective side lengths.

Outer Area =

(5s + 4)^2 = 25s^2 + 40s + 16

and Inner Area =

(3s - 2)^2 = 9s^2 - 12s + 4

.

The area of a square is equal to the square of its side length.

3

Subtract the inner garden area from the outer area to find the walkway area, distributing the negative sign to all terms of the inner area.

Walkway Area =

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 25s^2 + 40s + 16 - 9s^2 + 12s - 4 = 16s^2 + 52s + 12

square yards.

The area of the walkway is the difference between the total outer area and the inner garden area.

Key Concept

Operations on Polynomials

Question 9Question →

The polynomial $P(x)$ is defined by $P(x) = (2x^2 - 3x + 5)^2 - 4x(x^3 - 2x^2 - 7x + 1)$ . When $P(x)$ is written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 57

Answer

The coefficient of the

x^2

term is 57.

Expanding

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

yields

(2x^2 - 3x + 5)(2x^2 - 3x + 5) = 4x^4 - 12x^3 + 29x^2 - 30x + 25

. Distributing the

-4x

term yields

-4x(x^3 - 2x^2 - 7x + 1) = -4x^4 + 8x^3 + 28x^2 - 4x

. Combining the

x^2

terms from both expressions gives

29x^2 + 28x^2 = 57x^2

. Thus, the coefficient of the

x^2

term is 57.

Step-by-Step Solution

1

Expand the squared trinomial

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

4x^4 - 12x^3 + 29x^2 - 30x + 25

Expanding the first part of the expression by multiplying the trinomial by itself.

2

Distribute the term

-4x

to the trinomial

(x^3 - 2x^2 - 7x + 1)

-4x^4 + 8x^3 + 28x^2 - 4x

Expanding the second part of the polynomial expression while distributing the negative sign to all terms.

3

Combine the expanded expressions and isolate the

x^2

terms

29x^2 + 28x^2 = 57x^2

Adding the coefficients of the terms of degree 2 to find the combined coefficient.

Key Concept

Operations on Polynomials

Question 10Question →

A triangle has a base of $2x + 4$ inches and a height of $x - 3$ inches. What is the coefficient of $x$ when the expression representing the area of the triangle, in square inches, is written in standard form?

Show answer & explanation

Answer: -1

Answer

The coefficient of

x

is

-1

.

The area of a triangle is calculated using

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Substituting the given expressions, we get

\text{Area} = \frac{1}{2}(2x + 4)(x - 3)

. First, we can multiply the

\frac{1}{2}

by

(2x + 4)

, which simplifies to

x + 2

. Next, we expand

(x + 2)(x - 3)

using FOIL to get

x^2 - 3x + 2x - 6

. Combining like terms yields

x^2 - x - 6

. The coefficient of the

x

term in this simplified expression is

-1

.

Step-by-Step Solution

1

State the formula for the area of a triangle.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

To establish the mathematical relationship.

2

Substitute the given values into the formula.

\text{Area} = \frac{1}{2}(2x + 4)(x - 3)

To express the area in terms of the variable

x

.

3

Distribute the fraction to the first binomial.

x + 2

Distributing

\frac{1}{2}

to

(2x + 4)

simplifies the expression before multiplying.

4

Expand the product of the binomials.

x^2 - 3x + 2x - 6

Using the FOIL method to multiply

(x + 2)

and

(x - 3)

.

5

Combine the linear terms.

x^2 - x - 6

To simplify the polynomial and write it in standard form.

6

Identify the coefficient of

x

.

-1

The coefficient of the

x

term in

x^2 - x - 6

is

-1

.

Key Concept

Multiplying binomials and applying formulas in geometric contexts

Estimated Time:1m 0s

Question 11Question →

A square playground has a side length of $2x^3$ meters. A square sandbox with a side length of $x^3 - 2x$ meters is built inside the playground. Which of the following expressions represents the area, in square meters, of the playground that is NOT covered by the sandbox?

3x^6 - 4x^4 + 4x^2

3x^6 - 4x^2

3x^6 + 4x^4 + 4x^2

3x^6 + 4x^4 - 4x^2

3x^6 + 4x^3 - 4x^2

Show answer & explanation

Answer: $3x^6 + 4x^4 - 4x^2$

Answer

The correct expression is

3x^6 + 4x^4 - 4x^2

.

The expression

3x^6 + 4x^4 - 4x^2

is correct because the area of the playground is

(2x^3)^2 = 4x^6

and the area of the sandbox is

(x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2

. Subtracting the sandbox area from the playground area requires distributing the negative sign, resulting in

4x^6 - x^6 - (-4x^4) - 4x^2 = 3x^6 + 4x^4 - 4x^2

.

Step-by-Step Solution

1

Calculate the area of the square playground.

\text{Area}_{\text{playground}} = (2x^3)^2 = 4x^6

The area of a square is equal to the square of its side length, and applying the exponent rules yields

(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4x^6

.

2

Calculate the area of the square sandbox.

\text{Area}_{\text{sandbox}} = (x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2

Using the binomial squaring formula

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

, we square each term and compute the middle product, adding exponents for

x^3 \cdot x^1 = x^4

.

3

Subtract the sandbox's area from the playground's area.

4x^6 - (x^6 - 4x^4 + 4x^2) = 4x^6 - x^6 + 4x^4 - 4x^2 = 3x^6 + 4x^4 - 4x^2

Distribute the negative sign to all three terms inside the parentheses and combine the like terms of

x^6

.

Key Concept

Polynomial subtraction and squaring binomials with variables containing exponents.

Alternative Method

Evaluate the expression for a small integer value of

x

. If

x = 2

, the playground side length is

2(2)^3 = 16

, giving an area of

16^2 = 256

. The sandbox side length is

2^3 - 2(2) = 4

, giving an area of

4^2 = 16

. The remaining area is

256 - 16 = 240

. Substituting

x = 2

into the correct expression

3x^6 + 4x^4 - 4x^2

yields

3(64) + 4(16) - 4(4) = 192 + 64 - 16 = 240

, confirming its correctness.

Estimated Time:2m 0s

Question 12Question →

Two polynomial expressions are defined as $P(x) = (2x - 4)(x^2 - 5x + c)$ and $Q(x) = (x^2 + 3x - 4)(2x + a)$ , where $a$ and $c$ are constants. When $P(x)$ is expanded and simplified, it has no $x$ term. If the coefficient of the $x^2$ term in the expanded and simplified form of $Q(x)$ is equal to the coefficient of the $x^2$ term in the expanded and simplified form of $P(x)$ , what is the value of $a$ ?

Show answer & explanation

Answer: -20

Answer

The value of

a

is

-20

.

Expanding

P(x) = (2x - 4)(x^2 - 5x + c)

yields

2x^3 - 14x^2 + (2c + 20)x - 4c

. Since there is no

x

term,

2c + 20 = 0

, which means

c = -10

. This leaves the coefficient of the

x^2

term in

P(x)

as

-14

. Expanding

Q(x) = (x^2 + 3x - 4)(2x + a)

yields

2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

, so the coefficient of its

x^2

term is

a + 6

. Setting

a + 6 = -14

and solving for

a

gives

a = -20

.

Step-by-Step Solution

1

Expand the polynomial

P(x) = (2x - 4)(x^2 - 5x + c)

using the distributive property.

P(x) = 2x^3 - 14x^2 + (2c + 20)x - 4c

To identify the coefficients of each term in

P(x)

.

2

Set the coefficient of the

x

term,

2c + 20

, to

0

and solve for

c

.

c = -10

The problem states that

P(x)

has no

x

term when simplified, which means its coefficient must be zero.

3

Determine the coefficient of the

x^2

term in

P(x)

.

The coefficient of

x^2

is

-14

.

This coefficient will be equated to the

x^2

coefficient of

Q(x)

as per the problem constraints.

4

Expand the polynomial

Q(x) = (x^2 + 3x - 4)(2x + a)

using the distributive property.

Q(x) = 2x^3 + (a + 6)x^2 + (3a - 8)x - 4a

To identify the coefficient of the

x^2

term in

Q(x)

.

5

Set the coefficient of the

x^2

term in

Q(x)

, which is

a + 6

, equal to the coefficient of the

x^2

term in

P(x)

, which is

-14

, and solve for

a

.

a = -20

The problem states that these two coefficients are equal.

Key Concept

Operations on Polynomials

Estimated Time:2m 30s

Question 13Question →

A rectangle has a length of $y + 7$ inches and a width of $y - 3$ inches. Which of the following expressions represents the area, in square inches, of the rectangle?

y^2 - 21

y^2 + 4y + 21

y^2 + 4y - 21

y^2 - 4y - 21

5y - 21

Show answer & explanation

Answer: y^2 + 4y - 21

Answer

The expression

y^2 + 4y - 21

The area of a rectangle is found by multiplying its length by its width. The product of the dimensions is represented by the expression

(y + 7)(y - 3)

. Applying the distributive property gives

y^2 - 3y + 7y - 21

. Combining the like terms

-3y

and

7y

results in

+4y

. Therefore, the simplified expression for the area is

y^2 + 4y - 21

.

Step-by-Step Solution

1

Set up the area formula for the rectangle by multiplying its length and width.

Area

= (y + 7)(y - 3)

The area of a rectangle is calculated by multiplying its length by its width.

2

Use the distributive property to expand the product of the two binomials.

Area

= y(y) + y(-3) + 7(y) + 7(-3) = y^2 - 3y + 7y - 21

Each term in the first binomial must be multiplied by each term in the second binomial.

3

Combine the like terms to simplify the expression into standard form.

Area

= y^2 + 4y - 21

Combining

-3y

and

+7y

yields

+4y

.

Key Concept

Finding the area of a rectangle by multiplying binomial expressions using the distributive property.

Estimated Time:45s

Question 14Question →

A manufacturing company produces sheet metal. The area of a large rectangular sheet is represented by the expression $2x^2(3x^3 - 4x + 5)$ square inches. A square piece with a side length of $3x - 4$ inches is cut out from the sheet. Which of the following expressions represents the area of the remaining metal sheet, in square inches, for all $x > 2$ ?

6x^6 - 8x^3 + x^2 + 24x - 16

6x^5 - 8x^3 + x^2 - 16

6x^5 - 8x^3 + x^2 - 24x + 16

6x^5 - 8x^3 + x^2 + 24x - 16

6x^5 - 8x^3 + x^2 + 24x + 16

Show answer & explanation

Answer: $6x^5 - 8x^3 + x^2 + 24x - 16$

Answer

The remaining area of the metal sheet is represented by

6x^5 - 8x^3 + x^2 + 24x - 16

.

To find the remaining area, the area of the square piece must be subtracted from the area of the large rectangular sheet. Expanding the large rectangular sheet's area yields

2x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2

. Expanding the area of the square piece yields

(3x - 4)^2 = 9x^2 - 24x + 16

. Subtracting the second polynomial from the first requires distributing the negative sign to all terms:

(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16

. Combining the quadratic terms

10x^2 - 9x^2 = x^2

yields the correct simplified expression

6x^5 - 8x^3 + x^2 + 24x - 16

.

Step-by-Step Solution

1

Expand the expression for the area of the large rectangular sheet.

2x^2(3x^3 - 4x + 5) = 6x^5 - 8x^3 + 10x^2

Distribute the term

2x^2

to each term in the trinomial, using the exponent rule

x^a \cdot x^b = x^{a+b}

to combine the variable factors.

2

Expand the expression for the area of the square piece.

(3x - 4)^2 = 9x^2 - 24x + 16

Apply the binomial squaring formula

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

, where

a = 3x

and

b = 4

.

3

Subtract the area of the square piece from the area of the large rectangular sheet.

(6x^5 - 8x^3 + 10x^2) - (9x^2 - 24x + 16) = 6x^5 - 8x^3 + 10x^2 - 9x^2 + 24x - 16

Distribute the negative sign to each term of the subtracted polynomial, changing the sign of each term inside the second parenthesis.

4

Combine like terms and write the final expression in descending order.

6x^5 - 8x^3 + x^2 + 24x - 16

Combine the quadratic terms:

10x^2 - 9x^2 = x^2

. Arrange the remaining terms in descending order of their exponents.

Key Concept

Polynomial operations involving monomial distribution with exponent rules, binomial squaring, and polynomial subtraction with sign distribution.

Estimated Time:2m 0s

Question 15Question →

Let $P(x) = (3x^2 - 2x + 4)(2x - 5) - (4x^3 - 7x^2 + x - 3)$ . When $P(x)$ is simplified and written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: -12

Answer

The coefficient of the

x^2

term is -12.

Expanding

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

yields

6x^3 - 19x^2 + 18x - 20

. Distributing the negative sign across the second polynomial yields

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

. Combining the

x^2

terms gives

-19x^2 + 7x^2 = -12x^2

, so the coefficient of the

x^2

term is -12.

Step-by-Step Solution

1

Expand the product of the trinomial and the binomial:

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

6x^3 - 19x^2 + 18x - 20

To find the expanded form of the first polynomial component before subtraction

2

Distribute the negative sign across the second polynomial:

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

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

To prepare the second polynomial for combination of like terms

3

Combine the like terms from the two expanded components

2x^3 - 12x^2 + 17x - 17

To write the entire polynomial in standard form and identify the coefficient of

x^2

Key Concept

Operations on Polynomials

Question 16Question →

A closed cardboard box has a height of $x$ inches, a width of $3x - 2$ inches, and a length of $2x + 5$ inches. When the volume of the box, in cubic inches, is written as a polynomial in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 11

Answer

The coefficient of the

x^2

term is 11.

Expanding the volume expression

V(x) = x(3x - 2)(2x + 5)

yields

6x^3 + 11x^2 - 10x

. The coefficient of the

x^2

term is the numerical value associated with

x^2

, which is 11.

Step-by-Step Solution

1

Set up the polynomial expression for the volume.

V(x) = x(3x - 2)(2x + 5)

The volume of a rectangular prism is the product of its length, width, and height.

2

Multiply the binomials

(3x - 2)

and

(2x + 5)

by distributing terms.

(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

To find the product of two binomials, multiply each term of the first binomial by each term of the second binomial and combine like terms.

3

Distribute the monomial

x

to each term in the simplified trinomial.

x(6x^2 + 11x - 10) = 6x^3 + 11x^2 - 10x

The height

x

must scale the entire base area polynomial.

4

Identify the coefficient of the quadratic term

x^2

.

11

The coefficient of a term is the numerical factor multiplied by the variable part.

Key Concept

Multiplying polynomials and identifying coefficients of specific terms in the resulting standard form polynomial.

Question 17Question →

A company's weekly revenue, $R(x)$ , and weekly cost, $C(x)$ , in dollars, are modeled by the functions $R(x) = (2x^2 + 3)^2$ and $C(x) = x^2(x^3 - 5)$ , where $x$ represents the number of units produced and sold. Which of the following expressions represents the company's weekly profit, $P(x) = R(x) - C(x)$ ?

-x^5 + 4x^4 + 7x^2 + 9

-x^6 + 4x^4 + 17x^2 + 9

-x^5 + 4x^4 + 17x^2 + 9

-x^5 + 4x^4 + 5x^2 + 9

-x^6 + 4x^4 + 7x^2 + 9

Show answer & explanation

Answer: $-x^5 + 4x^4 + 17x^2 + 9$

Answer

The expression representing the weekly profit is

-x^5 + 4x^4 + 17x^2 + 9

.

To find the profit function, we subtract the cost function from the revenue function. First, expanding

(2x^2 + 3)^2

yields

4x^4 + 12x^2 + 9

. Second, distributing

x^2

over

(x^3 - 5)

yields

x^5 - 5x^2

. Subtracting these two functions gives

(4x^4 + 12x^2 + 9) - (x^5 - 5x^2)

. Distributing the subtraction sign changes the signs of the cost function to

-x^5 + 5x^2

. Combining like terms (

12x^2 + 5x^2 = 17x^2

) and organizing the expression in descending order yields

-x^5 + 4x^4 + 17x^2 + 9

.

Step-by-Step Solution

1

Expand the revenue function

R(x) = (2x^2 + 3)^2

using the binomial square identity

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

.

R(x) = (2x^2)^2 + 2(2x^2)(3) + 3^2 = 4x^4 + 12x^2 + 9

This simplifies the revenue expression into a standard polynomial form.

2

Expand the cost function

C(x) = x^2(x^3 - 5)

by distributing

x^2

to both terms inside the parentheses and applying the product rule for exponents

x^a \cdot x^b = x^{a+b}

.

C(x) = x^2(x^3) - x^2(5) = x^5 - 5x^2

This simplifies the cost expression into a standard polynomial form.

3

Subtract

C(x)

from

R(x)

by setting up

P(x) = R(x) - C(x)

and distributing the negative sign to all terms of

C(x)

.

P(x) = (4x^4 + 12x^2 + 9) - (x^5 - 5x^2) = 4x^4 + 12x^2 + 9 - x^5 + 5x^2

Profit is revenue minus cost, and distributing the negative sign correctly is essential.

4

Combine like terms and write the final expression in descending order of exponents.

P(x) = -x^5 + 4x^4 + 17x^2 + 9

To present the polynomial in standard form.

Key Concept

Polynomial operations, including expanding binomial squares, distributing monomial terms, and subtracting polynomials with careful attention to sign distribution.

Question 18Question →

An artist is designing a rectangular stained-glass window. The total area of the window, in square inches, is represented by the polynomial $8x^2 + 2x - 3$ . The area of the central blue glass section, in square inches, is represented by the product $(2x - 3)(x - 2)$ . The remaining portion of the window is made of clear glass. Which of the following expressions represents the area, in square inches, of the clear glass section?

6x^2 + 9x - 9

6x^2 - 5x + 3

6x^2 + 9x + 3

8x^2 + 7x - 9

6x^2 + 2x - 9

Show answer & explanation

Answer: $6x^2 + 9x - 9$

Answer

The expression representing the area of the clear glass section is

6x^2 + 9x - 9

.

The correct expression is

6x^2 + 9x - 9

. The area of the clear glass is found by expanding

(2x - 3)(x - 2)

to get

2x^2 - 7x + 6

and then subtracting this from the total area:

(8x^2 + 2x - 3) - (2x^2 - 7x + 6) = 8x^2 + 2x - 3 - 2x^2 + 7x - 6 = 6x^2 + 9x - 9

.

Step-by-Step Solution

1

Expand the product representing the area of the blue glass section using binomial multiplication.

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

Converting the factored area of the blue glass section to standard form is necessary before subtraction.

2

Set up the subtraction of the blue glass area from the total area.

(8x^2 + 2x - 3) - (2x^2 - 7x + 6)

The clear glass area is found by subtracting the blue glass area from the total window area.

3

Distribute the negative sign to each term of the second polynomial and combine like terms.

8x^2 + 2x - 3 - 2x^2 + 7x - 6 = (8x^2 - 2x^2) + (2x + 7x) + (-3 - 6) = 6x^2 + 9x - 9

Distributing the subtraction sign changes the signs of all terms inside the second set of parentheses, allowing for correct simplification.

Key Concept

Operations on Polynomials

Estimated Time:1m 30s

Question 19Question →

A landscape architect is designing a park layout. The total area of the park is represented by the polynomial $A(t) = (2t - 3)(3t^2 + 4t - 1)$ square meters, while the area allocated for a parking lot is represented by $P(t) = 2t^2(3t - 5)$ square meters, where $t$ represents a scaling factor. The remaining green space in the park is modeled by the polynomial $g(t) = A(t) - P(t)$ . When $g(t)$ is written in the standard form $at^3 + bt^2 + ct + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, what is the value of $b$ ?

Show answer & explanation

Answer: 9

Answer

The value of

b

, the coefficient of the

t^2

term in the standard form of

g(t)

, is

9

.

Expanding the polynomials gives

A(t) = 6t^3 - t^2 - 14t + 3

and

P(t) = 6t^3 - 10t^2

. Subtracting

P(t)

from

A(t)

yields

g(t) = (6t^3 - t^2 - 14t + 3) - (6t^3 - 10t^2) = 9t^2 - 14t + 3

. In the standard form

at^3 + bt^2 + ct + d

, the coefficient

b

of the

t^2

term is

9

.

Step-by-Step Solution

1

Expand the polynomial

A(t) = (2t - 3)(3t^2 + 4t - 1)

6t^3 - t^2 - 14t + 3

To represent the total area as a single polynomial in standard form before subtraction.

2

Expand the polynomial

P(t) = 2t^2(3t - 5)

6t^3 - 10t^2

To represent the parking lot area as a simplified polynomial in standard form.

3

Subtract

P(t)

from

A(t)

to find the green space polynomial

g(t)

9t^2 - 14t + 3

Subtracting

P(t)

from

A(t)

requires distributing the negative sign to both terms, yielding

-t^2 - (-10t^2) = 9t^2

.

4

Identify the coefficient

b

of the

t^2

term in

at^3 + bt^2 + ct + d

9

Comparing

g(t) = 9t^2 - 14t + 3

to the standard cubic form reveals that

a = 0

and

b = 9

.

Key Concept

Polynomial operations including expansion of products and subtraction with negative sign distribution

Question 20Question →

The volume of a water tank, in cubic feet, is given by the polynomial $V(t) = 3t(2t - 1)(t + 4)$ , where $t$ represents the time in hours since the pump was turned on. When the polynomial is written in standard form, what is the coefficient of the $t^2$ term?

Show answer & explanation

Answer: 21

Answer

21

To find the coefficient of the

t^2

term, we perform polynomial multiplication on the expression

V(t) = 3t(2t - 1)(t + 4)

. First, multiplying the binomials

(2t - 1)(t + 4)

yields

2t^2 + 7t - 4

. Next, distributing the

3t

term to the trinomial yields

6t^3 + 21t^2 - 12t

. The coefficient of the

t^2

term in this standard form polynomial is 21.

Step-by-Step Solution

1

Expand the product of the two binomials

(2t - 1)(t + 4)

.

2t^2 + 7t - 4

Use the distributive property (FOIL) to multiply the binomials:

2t(t) + 2t(4) - 1(t) - 1(4) = 2t^2 + 8t - t - 4 = 2t^2 + 7t - 4

.

2

Multiply the resulting trinomial by the monomial

3t

.

6t^3 + 21t^2 - 12t

Distribute

3t

to each term:

3t(2t^2) + 3t(7t) + 3t(-4) = 6t^3 + 21t^2 - 12t

.

3

Identify the coefficient of the

t^2

term.

21

The coefficient is the numerical factor of the

t^2

term, which is 21.

Key Concept

Operations on Polynomials

Operations on Polynomials

Answer

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