Factoring Polynomials Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

Question 21Question →

Consider the polynomial $8x^2 - 10x - 3$ . When this expression is factored completely into the form $(4x + a)(2x - b)$ , where $a$ and $b$ are positive integers, what are the corresponding values for $a$ and $b$ ?

Fill in the blanks below

The value of

a

is and the value of

b

is .

Show answer & explanation

Answer

The value of a is 1 and the value of b is 3.

By applying the AC method, the quadratic

8x^2 - 10x - 3

can be rewritten as

8x^2 - 12x + 2x - 3

and factored by grouping into

4x(2x - 3) + 1(2x - 3) = (4x + 1)(2x - 3)

. Aligning this with

(4x + a)(2x - b)

directly shows that

a = 1

and

b = 3

.

Step-by-Step Solution

1

Identify the type of polynomial and the appropriate factoring method.

The polynomial

8x^2 - 10x - 3

is a non-monic quadratic trinomial (where the leading coefficient is not 1). We can use the ac-method.

The ac-method involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

2

Calculate the ac product and find the target sum.

The product is

a \times c = 8 \times (-3) = -24

. The target sum is the middle coefficient, which is

-10

.

These values are required to split the middle term for factoring by grouping.

3

Find two integers that multiply to -24 and add to -10.

The numbers

-12

and

2

meet these conditions, since

(-12) \times 2 = -24

and

(-12) + 2 = -10

.

This allows us to rewrite the middle term

-10x

without changing the value of the polynomial.

4

Rewrite the middle term and factor by grouping.

8x^2 - 12x + 2x - 3

. Grouping the terms yields

(8x^2 - 12x) + (2x - 3)

. Factoring out the greatest common factor from each group gives

4x(2x - 3) + 1(2x - 3)

.

Grouping transforms the four-term expression into a format where a common binomial factor can be extracted.

5

Factor out the common binomial factor and determine the final values.

Factoring out

(2x - 3)

leaves

(4x + 1)(2x - 3)

. Comparing this to the given form

(4x + a)(2x - b)

reveals that

a = 1

and

b = 3

.

Matching the derived factorization with the required format identifies the specific constants requested by the problem.

Key Concept

Factoring non-monic quadratic trinomials using the AC method and grouping.

Question 22Question →

Find the completely factored form of the given polynomial:

-12x^2 + 30x + 72

Which of the following options shows the correct factorization?

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

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

-2(2x + 3)(3x - 12)

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

Show answer & explanation

Answer: $-6(2x + 3)(x - 4)$

Answer

The completely factored form is

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

.

The polynomial requires factoring out the Greatest Common Factor (GCF) of -6, which leaves the trinomial

2x^2 - 5x - 12

. Factoring this trinomial yields the binomials

(2x + 3)

and

(x - 4)

. Multiplying these by the GCF gives the fully factored expression.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the polynomial

-12x^2 + 30x + 72

.

The GCF is -6. Factoring this out gives:

-6(2x^2 - 5x - 12)

.

Factoring out a negative GCF makes the leading coefficient of the trinomial positive, which simplifies the subsequent factoring steps.

2

Factor the remaining quadratic trinomial

2x^2 - 5x - 12

by finding two integers that multiply to

(2 \times -12) = -24

and add to the middle coefficient

-5

.

The two integers are -8 and 3.

These integers will be used to split the middle term to factor by grouping.

3

Rewrite the trinomial by splitting the middle term using the integers found, then factor by grouping.

2x^2 - 8x + 3x - 12 = 2x(x - 4) + 3(x - 4) = (2x + 3)(x - 4)

.

Grouping allows us to factor out common binomials to find the factored form of the trinomial.

4

Combine the GCF with the factored binomials to write the final completely factored expression.

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

A completely factored polynomial must include all extracted common factors alongside the irreducible binomial components.

Key Concept

Factoring polynomials using a Greatest Common Factor (GCF) and factoring non-monic trinomials.

Question 23Question →

Which of the following expressions is equivalent to $6x^2 - 15x - 36$ and is factored completely?

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

(2x + 3)(3x - 12)

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

3(2x + 1)(x - 12)

Show answer & explanation

Answer: $3(2x + 3)(x - 4)$

Answer

The completely factored expression is

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

.

The correct answer identifies the GCF of 3 and correctly factors the remaining quadratic

2x^2 - 5x - 12

into

(2x + 3)(x - 4)

. Expanding this result confirms it equals

6x^2 - 15x - 36

and no further factors can be extracted.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all three terms.

3(2x^2 - 5x - 12)

The coefficients 6, -15, and -36 are all divisible by 3.

2

Find two numbers that multiply to

ac

(where

a=2

and

c=-12

, so

2 \times -12 = -24

) and add to the middle coefficient (

b=-5

).

The numbers are

-8

and

3

.

(-8) \times 3 = -24

and

(-8) + 3 = -5

.

3

Rewrite the trinomial by splitting the middle term using these two numbers.

3(2x^2 - 8x + 3x - 12)

This sets up the polynomial for factoring by grouping.

4

Factor the trinomial by grouping.

3[2x(x - 4) + 3(x - 4)] = 3(2x + 3)(x - 4)

Factoring out

(x - 4)

from both groups provides the final completely factored form.

Key Concept

Factoring out a GCF followed by factoring a non-monic trinomial.

Question 24Question →

Which of the following represents the completely factored form of the algebraic expression $30v^3 - 35v^2 - 15v$ ?

5v(3v + 1)(2v - 3)

5v(3v - 1)(2v + 3)

v(15v + 5)(2v - 3)

5v(v - 9)(v + 2)

Show answer & explanation

Answer: $5v(3v + 1)(2v - 3)$

Answer

The completely factored form is

5v(3v + 1)(2v - 3)

.

The correct answer accurately extracts the Greatest Common Factor of 5v from the original expression, leaving the non-monic trinomial 6v^2 - 7v - 3. Factoring this remaining trinomial by grouping correctly yields the binomials (3v + 1) and (2v - 3). Expanding these prime factors back confirms they produce the exact original expression.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all three terms in the polynomial

30v^3 - 35v^2 - 15v

.

5v(6v^2 - 7v - 3)

Factoring out the GCF simplifies the remaining polynomial and is a necessary first step for complete factorization.

2

Analyze the remaining quadratic trinomial

6v^2 - 7v - 3

to factor it by grouping. Find two numbers that multiply to

a \times c

(

6 \times -3 = -18

) and add to

b

(

-7

).

The two numbers are

-9

and

2

.

These numbers allow us to split the middle term (

-7v

) into two separate terms to enable factoring by grouping.

3

Rewrite the trinomial by splitting the middle term using the numbers found, then factor by grouping.

6v^2 - 9v + 2v - 3 = 3v(2v - 3) + 1(2v - 3) = (3v + 1)(2v - 3)

Grouping the terms reveals the common binomial factor of

(2v - 3)

.

4

Combine the GCF extracted in step 1 with the binomial factors found in step 3.

5v(3v + 1)(2v - 3)

This represents the original expression as a product of its simplest irreducible factors.

Key Concept

Factoring a polynomial by first extracting the greatest common factor (GCF), followed by factoring a non-monic quadratic trinomial using grouping.

Question 25Question →

Decompose the quadratic trinomial $6x^2 - 11x - 10$ into a product of two binomial factors structured as $(ax - b)(cx + d)$ . Assume all parameters ( $a$ , $b$ , $c$ , and $d$ ) are strictly positive integers. To ensure a unique arrangement, the leading coefficient $a$ must be strictly less than $c$ . Provide the values of the four parameters.

Fill in the blanks below

a =

,

b =

,

c =

, and

d =

.

Show answer & explanation

Answer

The parameters are

a = 2

,

b = 5

,

c = 3

, and

d = 2

.

Factoring the polynomial

6x^2 - 11x - 10

using the AC method produces

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

. Matching these two factors against the template

(ax - b)(cx + d)

under the rule that

a < c

(since

2 < 3

) perfectly determines the positive integers as

a = 2

,

b = 5

,

c = 3

, and

d = 2

.

Step-by-Step Solution

1

Identify the factoring technique for a non-monic quadratic trinomial in the form

Ax^2 + Bx + C

.

The AC method (factoring by grouping) is required because the leading coefficient

A = 6

is greater than 1.

Directly finding factors of the constant

C

that sum to

B

only works when

A = 1

.

2

Multiply the leading coefficient

A

by the constant term

C

.

6 \times (-10) = -60

.

This establishes the target product needed to properly split the middle term for grouping.

3

Find two integers that multiply to

-60

and add to the middle coefficient

B = -11

.

The correct pair of integers is

-15

and

4

.

(-15) \times 4 = -60

and

-15 + 4 = -11

.

4

Rewrite the middle term using these two integers and factor the resulting four-term expression by grouping.

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

.

Grouping the first two terms and the last two terms reveals a common binomial factor of

(2x - 5)

.

5

Match the factored expression to the required template

(ax - b)(cx + d)

while adhering to the condition

a < c

.

The factors are

(2x - 5)

and

(3x + 2)

. Since

2 < 3

,

(ax - b)

corresponds exactly to

(2x - 5)

, yielding

a=2

and

b=5

. Consequently,

(cx + d)

corresponds to

(3x + 2)

, yielding

c=3

and

d=2

.

This guarantees a unique, correctly-ordered set of strictly positive parameters.

Key Concept

Factoring non-monic quadratic trinomials via grouping

Question 26Question →

Complete the factorization of the perfect square trinomial below by filling in the missing number.

Fill in the blanks below

$x^2 + 14x + 49 = (x + )^2$

Show answer & explanation

Answer

7

A perfect square trinomial takes the general form

x^2 + 2bx + b^2 = (x + b)^2

. In the given expression

x^2 + 14x + 49

, the constant term is

49

, which is

7^2

. The middle term is

14x

, which is equal to

2 \times 7 \times x

. Therefore, the expression factors directly as

(x + 7)^2

, meaning the missing number in the blank is 7.

Step-by-Step Solution

1

Recognize the polynomial form

The expression

x^2 + 14x + 49

matches the structure of a perfect square trinomial:

a^2 + 2ab + b^2

.

The first term is a perfect square (

x^2

), and the last term is a perfect square (

49 = 7^2

).

2

Verify the middle term

Checking the middle term:

2 \times x \times 7 = 14x

. This matches the middle term of the given expression.

Confirming the middle term proves the expression can be factored exactly as

(a + b)^2

.

3

Determine the missing value

The expression factors as

(x + 7)^2

.

Using the formula

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

where

a = x

and

b = 7

.

Key Concept

Factoring perfect square trinomials

Question 27Question →

An engineer needs to simplify a calculation involving the volume of a component, modeled by the polynomial $4y^3 + 12y^2 - 72y$ . What is the fully factored equivalent of this polynomial?

4y(y + 6)(y - 3)

4y(y - 6)(y + 3)

4y(y^2 + 3y - 18)

4y(y + 6)(y + 3)

Show answer & explanation

Answer: $4y(y + 6)(y - 3)$

Answer

4y(y + 6)(y - 3)

To fully factor the given expression, first extract the greatest common factor (GCF). All terms share a common factor of

4y

, yielding

4y(y^2 + 3y - 18)

. Next, factor the resulting quadratic trinomial

y^2 + 3y - 18

. The two numbers that multiply to

-18

and sum to

3

are

6

and

-3

. This gives the binomial factors

(y + 6)

and

(y - 3)

. Combining these with the GCF results in the completely factored form

4y(y + 6)(y - 3)

.

Step-by-Step Solution

1

Identify and extract the greatest common factor (GCF) from all terms in the polynomial

4y^3 + 12y^2 - 72y

.

4y(y^2 + 3y - 18)

Factoring out the GCF simplifies the remaining polynomial, making it easier to factor completely.

2

Factor the remaining quadratic trinomial

y^2 + 3y - 18

by finding two integers that multiply to

-18

and add to

3

.

(y + 6)(y - 3)

The integers

6

and

-3

satisfy both conditions:

6 \times (-3) = -18

and

6 + (-3) = 3

.

3

Combine the GCF with the factored binomials to state the final expression.

4y(y + 6)(y - 3)

This represents the original polynomial decomposed entirely into its lowest degree factors.

Key Concept

Factoring Polynomials with a GCF

Estimated Time:1m 0s

Question 28Question →

Pair each algebraic expression with its completely factored equivalent.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

4y^2 - 16y

matches with

4y(y - 4)

;

y^2 - 16

matches with

(y - 4)(y + 4)

;

y^2 - 8y + 16

matches with

(y - 4)^2

; and

4y + 16

matches with

4(y + 4)

.

Each polynomial requires a distinct factoring method. Identifying the correct structure—such as GCF, difference of squares, or perfect square trinomial—leads directly to the completely factored form.

Step-by-Step Solution

1

Factor

4y^2 - 16y

4y(y - 4)

The terms share a greatest common factor of

4y

.

2

Factor

y^2 - 16

(y - 4)(y + 4)

It follows the difference of squares pattern

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

.

3

Factor

y^2 - 8y + 16

(y - 4)^2

It is a perfect square trinomial, as

(-4) \times (-4) = 16

and

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

.

4

Factor

4y + 16

4(y + 4)

The terms share a greatest common factor of

4

.

Key Concept

Factoring polynomials using greatest common factors, difference of squares, and perfect square trinomial patterns.

Question 29Question →

Match each polynomial expression to its completely factored form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each polynomial with its mathematically equivalent fully factored expression.

Each polynomial requires identifying its specific factoring archetype: the difference of two squares, a standard quadratic trinomial, GCF extraction followed by recognizing a perfect square trinomial, or factoring a non-monic trinomial.

Step-by-Step Solution

1

Analyze the binomial

x^2 - 16

.

Matches to

(x - 4)(x + 4)

.

It represents a difference of perfect squares.

2

Analyze the trinomial

x^2 + 6x + 8

.

Matches to

(x + 2)(x + 4)

.

The constants 2 and 4 have a product of 8 and a sum of 6.

3

Analyze the polynomial

2x^2 - 12x + 18

.

Matches to

2(x - 3)^2

.

Factoring out the common factor of 2 leaves

x^2 - 6x + 9

, which is the square of

(x - 3)

.

4

Analyze the polynomial

3x^2 - 5x - 2

.

Matches to

(3x + 1)(x - 2)

.

Multiplying the binomials

(3x + 1)

and

(x - 2)

confirms it expands to the original expression.

Key Concept

Factoring different archetypes of polynomials, including the difference of squares, monic trinomials, non-monic trinomials, and extracting the Greatest Common Factor (GCF).

Question 30Question →

To optimize a packaging design, a technician models the surface area using the algebraic expression $18xy^2 - 24xy - 10x$ . What is the fully factored equivalent of this expression?

2x(3y - 5)(3y + 1)

2x(3y + 5)(3y - 1)

x(6y - 10)(3y + 1)

2xy(9y - 17)

Show answer & explanation

Answer: $2x(3y - 5)(3y + 1)$

Answer

The fully factored equivalent is

2x(3y - 5)(3y + 1)

.

The expression is fully factored by first extracting the greatest common factor of

2x

, resulting in

2x(9y^2 - 12y - 5)

. Then, using the ac-method, the remaining non-monic quadratic trinomial is factored into

(3y - 5)(3y + 1)

, yielding the final correct product.

Step-by-Step Solution

1

Identify and extract the greatest common factor (GCF) from all three terms.

2x(9y^2 - 12y - 5)

Extracting the GCF simplifies the polynomial and prepares the remaining quadratic trinomial for factoring.

2

Identify the components of the remaining quadratic trinomial

ay^2 + by + c

.

Here,

a = 9

,

b = -12

, and

c = -5

.

This establishes the values needed to use the ac-method for factoring non-monic quadratics.

3

Find two numbers that multiply to

ac

and add to

b

.

ac = 9 \times -5 = -45

. The two numbers that multiply to

-45

and add to

-12

are

-15

and

3

.

These numbers allow the middle term to be split, enabling factoring by grouping.

4

Rewrite the middle term and factor the trinomial by grouping.

9y^2 - 15y + 3y - 5 = 3y(3y - 5) + 1(3y - 5) = (3y - 5)(3y + 1)

This completes the factorization of the trinomial portion of the expression.

5

Combine the originally extracted GCF with the factored binomials.

2x(3y - 5)(3y + 1)

This yields the complete and fully factored form of the original polynomial.

Key Concept

Factoring Polynomials with a GCF

Question 31Question →

A rectangular garden has an area given by the expression $4n^2 - 4n - 24$ . Which of the following represents the completely factored form of this area expression?

4(n - 3)(n + 2)

4(n + 3)(n - 2)

(4n - 12)(n + 2)

4(n - 6)(n + 1)

Show answer & explanation

Answer: $4(n - 3)(n + 2)$

Answer

The completely factored form is

4(n - 3)(n + 2)

.

To factor the expression completely, first remove the greatest common factor of 4 from all terms, which gives

4(n^2 - n - 6)

. Then, factor the trinomial inside the parentheses. We need two numbers that multiply to -6 and add to -1, which are -3 and 2. This gives the completely factored form

4(n - 3)(n + 2)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all terms in the polynomial

4n^2 - 4n - 24

.

4(n^2 - n - 6)

Factoring out the GCF simplifies the polynomial and is the required first step for complete factorization.

2

Factor the remaining quadratic trinomial

n^2 - n - 6

by finding two numbers that multiply to -6 and add to -1.

The numbers are -3 and 2, yielding

(n - 3)(n + 2)

.

This breaks down the trinomial into its constituent binomial factors.

3

Combine the GCF and the binomial factors to write the final expression.

4(n - 3)(n + 2)

This represents the full, completely factored form of the original area expression.

Key Concept

Factoring out a GCF followed by factoring a quadratic trinomial

Question 32Question →

Which of the following expressions represents the completely factored form of the polynomial $3k^3 - 15k^2 - 42k$ ?

3k(k + 7)(k - 2)

k(3k - 21)(k + 2)

3k(k - 7)(k - 2)

3k(k - 7)(k + 2)

Show answer & explanation

Answer: $3k(k - 7)(k + 2)$

Answer

The completely factored form is

3k(k - 7)(k + 2)

.

The polynomial is completely factored by first extracting the Greatest Common Factor,

3k

, which simplifies the expression to

3k(k^2 - 5k - 14)

. Next, the remaining trinomial is factored by finding two numbers that multiply to

-14

and add to

-5

. These numbers are

-7

and

2

. Combining these gives the final fully factored form of

3k(k - 7)(k + 2)

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial.

The GCF of

3k^3

,

-15k^2

, and

-42k

is

3k

. Factoring this out leaves

3k(k^2 - 5k - 14)

.

Factoring out the GCF simplifies the remaining polynomial and is the required first step for complete factorization.

2

Factor the remaining quadratic trinomial,

k^2 - 5k - 14

.

Find two integers that multiply to

-14

(the constant term) and add to

-5

(the coefficient of the middle term). These integers are

-7

and

2

. This yields the binomial factors

(k - 7)

and

(k + 2)

.

These binomials represent the linear factors of the quadratic expression.

3

Combine the GCF and the binomial factors to write the fully factored expression.

3k(k - 7)(k + 2)

This represents the original polynomial as a product of its simplest prime factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Estimated Time:1m 0s

Question 33Question →

The algebraic expression $5x^2 + 35x$ can be completely factored into the form $5x(x + c)$ . What is the value of the constant $c$ ?

Show answer & explanation

Answer: 7; +7; seven; Seven

Answer

7

By extracting the greatest common factor of

5x

from

5x^2 + 35x

, we divide both terms by

5x

. This leaves

(x + 7)

inside the parentheses. Comparing this to the given structure

5x(x + c)

, it is clear that

c

must be 7.

Step-by-Step Solution

1

Identify the greatest common factor (GCF) of the terms

5x^2

and

35x

.

The GCF is

5x

.

Both terms are divisible by 5 and share a minimum of one

x

.

2

Divide each term of the original expression by the GCF to find the binomial factor.

\frac{5x^2}{5x} = x

and

\frac{35x}{5x} = 7

.

Factoring out the GCF is the reverse of distributing it.

3

Write the factored form and compare it to the given form

5x(x + c)

.

The factored form is

5x(x + 7)

. Therefore, comparing it to

5x(x + c)

,

c = 7

.

Matching corresponding terms reveals the value of the unknown constant.

Key Concept

Factoring the Greatest Common Factor (GCF)

Question 34Question →

When the polynomial $5x^2 - 20$ is factored completely into the form $k(x - c)(x + c)$ , where $k$ and $c$ are positive integers, what is the value of the sum $k + c$ ?

Show answer & explanation

Answer: 7; seven

Answer

7

The polynomial

5x^2 - 20

can be factored completely by first pulling out the greatest common factor,

5

, leaving

5(x^2 - 4)

. The expression inside the parentheses is a difference of squares and factors into

(x - 2)(x + 2)

. This makes the completely factored form

5(x - 2)(x + 2)

. Comparing this to the form

k(x - c)(x + c)

, we see that

k = 5

and

c = 2

. Their sum is

5 + 2 = 7

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from the polynomial.

5x^2 - 20 = 5(x^2 - 4)

Both terms share a common numerical factor of

5

. Factoring it out simplifies the remaining polynomial.

2

Factor the remaining binomial as a difference of squares.

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

The binomial

x^2 - 4

follows the pattern

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

, where

a = x

and

b = 2

.

3

Identify the values of

k

and

c

from the given form.

k = 5

and

c = 2

By matching

5(x - 2)(x + 2)

to the target form

k(x - c)(x + c)

, the constant multiplier is

5

and the positive integer in the binomial is

2

.

4

Calculate the sum of

k

and

c

.

5 + 2 = 7

The question asks for the final value of the sum

k + c

.

Key Concept

Factoring polynomials using the greatest common factor and the difference of squares.

Question 35Question →

To determine the critical points of a trajectory, a physicist needs to fully factor the quadratic expression $4x^2 - 8x - 60$ . What is the completely factored equivalent of this expression?

4(x - 5)(x + 3)

4(x + 5)(x - 3)

(4x - 20)(x + 3)

4(x - 5)(x - 3)

Show answer & explanation

Answer: $4(x - 5)(x + 3)$

Answer

The completely factored form of the expression is

4(x - 5)(x + 3)

.

The correct answer is found by first extracting the Greatest Common Factor (GCF), which is

4

, leaving

4(x^2 - 2x - 15)

. Then, the inner trinomial

x^2 - 2x - 15

is factored into

(x - 5)(x + 3)

because

-5

and

3

multiply to

-15

and add to

-2

. Combining these gives the fully factored form

4(x - 5)(x + 3)

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression

4x^2 - 8x - 60

.

The GCF of

4

,

-8

, and

-60

is

4

. Factoring it out yields

4(x^2 - 2x - 15)

.

Extracting the GCF first simplifies the remaining polynomial, making the trinomial easier to factor.

2

Factor the remaining quadratic trinomial

x^2 - 2x - 15

by finding two numbers that multiply to the constant term (

-15

) and add to the middle term's coefficient (

-2

).

The numbers are

-5

and

3

, so the trinomial factors to

(x - 5)(x + 3)

.

Expanding

(x - 5)(x + 3)

gives

x^2 + 3x - 5x - 15

, which simplifies back to

x^2 - 2x - 15

.

3

Combine the GCF and the factored binomials to write the completely factored form.

4(x - 5)(x + 3)

This represents the original expression broken down into its prime polynomial factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Estimated Time:1m 0s

Question 36Question →

Complete the factored form of the polynomial by finding the missing values.

Fill in the blanks below

The polynomial

6x^2 - 6x - 72

can be factored completely into the form

6(x - a)(x + b)

. The missing positive integers are

a =

and

b =

.

Show answer & explanation

Answer

The missing values are a = 4 and b = 3. The completely factored form is 6(x - 4)(x + 3).

To factor the polynomial

6x^2 - 6x - 72

completely, we first factor out the greatest common factor (GCF), which is

6

. This leaves us with the expression

6(x^2 - x - 12)

. Next, we factor the quadratic trinomial

x^2 - x - 12

. We need to find two integers that multiply to give

-12

and add up to

-1

. Those integers are

-4

and

3

. This means the trinomial factors as

(x - 4)(x + 3)

. Combining these, the complete factorization is

6(x - 4)(x + 3)

. By comparing this result to the given format

6(x - a)(x + b)

, we can see that the first missing value is

a = 4

and the second missing value is

b = 3

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from the polynomial.

The GCF of the terms

6x^2

,

-6x

, and

-72

is

6

. Factoring this out gives:

6(x^2 - x - 12)

.

Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor into binomials.

2

Factor the remaining quadratic trinomial

x^2 - x - 12

.

We look for two numbers that multiply to

-12

and add up to

-1

(the coefficient of the middle term). These numbers are

-4

and

3

. Thus,

x^2 - x - 12

factors to

(x - 4)(x + 3)

.

A monic quadratic trinomial

x^2 + px + q

factors into

(x + m)(x + n)

where

m \times n = q

and

m + n = p

.

3

Write the complete factorization and match it to the given format.

The completely factored expression is

6(x - 4)(x + 3)

. Matching this to the given form

6(x - a)(x + b)

, we determine that

a = 4

and

b = 3

.

This directly identifies the specific positive integers required to complete the given algebraic expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a standard quadratic trinomial.

Question 37Question →

During an algebra calculation, an engineer needs to rewrite the expression $2x^2 + 7x - 15$ in its completely factored form. Which of the following is equivalent to this expression?

(2x - 3)(x + 5)

(2x + 3)(x - 5)

(2x - 5)(x + 3)

(2x + 15)(x - 1)

Show answer & explanation

Answer: $(2x - 3)(x + 5)$

Answer

The correct factorization is

(2x - 3)(x + 5)

.

The correct answer is mathematically equivalent to the original polynomial. Expanding it using the FOIL method confirms this:

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

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic expression

ax^2 + bx + c

.

a = 2

,

b = 7

, and

c = -15

.

Identifying these values is required to determine the target product and sum for factoring by grouping.

2

Find two numbers that multiply to

a \times c

and add up to

b

.

The numbers are

10

and

-3

because

10 \times (-3) = -30

and

10 + (-3) = 7

.

These specific numbers allow us to logically split the middle term of the polynomial.

3

Rewrite the middle term

7x

using these two numbers and factor the polynomial by grouping.

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

.

Extracting the greatest common factor from each pair reveals the common binomial factor, completing the factorization.

Key Concept

Factoring non-monic quadratic trinomials by grouping

Question 38Question →

Match each polynomial expression with its completely factored algebraic form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expressions correctly match their completely factored forms by applying the rules for difference of squares, perfect trinomials, extracting a greatest common factor, and factoring by grouping.

Each polynomial must be analyzed to find its unique factors. Recognizing perfect squares, extracting common factors first, and finding the correct binomial number pairs based on sum and product rules lead to the correct matches.

Step-by-Step Solution

1

Identify the structure of each polynomial to determine the correct factoring technique.

Categorize the expressions into difference of squares, monic trinomials, GCF + trinomial, and non-monic trinomials.

Applying the correct initial technique is essential to fully factoring the polynomial.

2

Apply the corresponding factoring method to break down the polynomial into its most basic prime factors.

Each expression is rewritten as a product of binomials and, if applicable, a constant scalar.

This verifies that the polynomial is factored completely, revealing the correct match.

Key Concept

Factoring Polynomials

Question 39Question →

To analyze the zeroes of a mathematical function, the algebraic expression $3x^3 + 6x^2 - 9x$ must be written as a product of its simplest linear factors. Which of the following represents the completely factored equivalent of this expression?

3x(x - 3)(x + 1)

x(3x + 9)(x - 1)

3x(x + 3)(x - 1)

3x^2(x - 1)

Show answer & explanation

Answer: $3x(x + 3)(x - 1)$

Answer

The completely factored equivalent is

3x(x + 3)(x - 1)

.

The correct answer successfully extracts the greatest common factor,

3x

, leaving the quadratic trinomial

x^2 + 2x - 3

. This trinomial factors correctly into

(x + 3)(x - 1)

because

3

and

-1

multiply to

-3

and add to positive

2

.

Step-by-Step Solution

1

Identify and extract the Greatest Common Factor (GCF) from all three terms in the expression

3x^3 + 6x^2 - 9x

.

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

Factoring out the GCF simplifies the remaining polynomial, making it possible to factor the quadratic trinomial.

2

Find two integers that multiply to the constant term (

-3

) and add to the middle coefficient (

2

).

The correct integers are

3

and

-1

.

These values are needed to factor the quadratic trinomial

x^2 + 2x - 3

into two binomials.

3

Rewrite the trinomial as a product of two binomials and combine it with the previously extracted GCF.

3x(x + 3)(x - 1)

This provides the final, completely factored form of the original algebraic expression.

Key Concept

Factoring Polynomials via GCF and Trinomial Methods

Question 40Question →

An architect is designing a rectangular layout where the area is represented by the algebraic expression $4y^2 + 12y - 40$ . To determine the possible dimensions of the layout in terms of $y$ , the architect needs to rewrite this expression in its completely factored form. Which of the following represents the completely factored expression?

4(y + 5)(y - 2)

4(y - 5)(y + 2)

(4y - 8)(y + 5)

4(y + 5)(y + 2)

Show answer & explanation

Answer: $4(y + 5)(y - 2)$

Answer

The completely factored expression is

4(y + 5)(y - 2)

.

The correct answer completely factors out the GCF of 4 from all terms and correctly factors the remaining trinomial

y^2 + 3y - 10

into

(y + 5)(y - 2)

.

Step-by-Step Solution

1

Identify and extract the greatest common factor (GCF) from all terms in the expression

4y^2 + 12y - 40

.

4(y^2 + 3y - 10)

Factoring out the GCF simplifies the polynomial and makes the remaining trinomial easier to factor.

2

Find two integers that multiply to the constant term (-10) and add to the coefficient of the middle term (3).

The correct integers are 5 and -2.

These integers are required to construct the binomial factors of the simplified trinomial

y^2 + 3y - 10

.

3

Write the trinomial as a product of two binomials using the integers found, and combine them with the GCF.

4(y + 5)(y - 2)

This represents the original expression fully broken down into its prime polynomial factors.

Key Concept

Factoring Polynomials by Extracting the GCF and Grouping

Factoring Polynomials

Answer

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Key Concept

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