ACCUPLACER Advanced Algebra & Functions Quiz: Factoring Gcf

20 questions · exam conditions

0:00

Factoring GcfQuestion 1 of 20

The expression $45r^6s^3t^2 - 30r^4s^5t + 75r^5s^2t^4$ can be factored as $15r^4s^2t \cdot Q$ . What is $Q$ ?

9r^2st - 6s^3 + 15rt^3

3r^2st - 2s^3t + 5rt^3

45r^2st - 30s^3 + 75rt^3

3r^2st - 2s^3 + 5rt^3

← Back to quizzes

ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Factoring Gcf

Practice Factoring Gcf in ACCUPLACER Advanced Algebra & Functions with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Factoring Gcf, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The expression $45r^6s^3t^2 - 30r^4s^5t + 75r^5s^2t^4$ can be factored as $15r^4s^2t \cdot Q$ . What is $Q$ ?

$9r^2st - 6s^3 + 15rt^3$
$3r^2st - 2s^3t + 5rt^3$
$45r^2st - 30s^3 + 75rt^3$
$3r^2st - 2s^3 + 5rt^3$ (correct answer)

Explanation: When you see a polynomial expression that needs to be factored with a given factor, you're working with the distributive property in reverse. The key is to divide each term of the original expression by the given factor to find what remains. To find

Q

, you need to divide each term of

45r^6s^3t^2 - 30r^4s^5t + 75r^5s^2t^4

15r^4s^2t

: First term:

\frac{45r^6s^3t^2}{15r^4s^2t} = 3r^2st

Second term:

\frac{-30r^4s^5t}{15r^4s^2t} = -2s^3

Third term:

\frac{75r^5s^2t^4}{15r^4s^2t} = 5rt^3

Therefore,

Q = 3r^2st - 2s^3 + 5rt^3

, which is choice D. Choice A contains

9r^2st

instead of

3r^2st

in the first term, suggesting an error in dividing coefficients (45 ÷ 15 = 3, not 9). Choice B has the correct first and third terms but incorrectly shows

-2s^3t

instead of

-2s^3

in the middle term—this represents keeping an extra factor of

t

that should have been divided out. Choice C shows

45r^2st

and

75rt^3

, indicating the coefficients weren't divided at all, and the middle term

-30s^3

is missing the

t

factor entirely. When factoring polynomials on the Accuplacer, always verify your answer by multiplying the factors back together. This catches division errors with coefficients or variables and ensures you haven't missed any terms in the factorization process.

Question 2

The expression $8x^2y^3 - 12x^3y^2$ is factored into the form $A \cdot B$ , where $A$ is the greatest common monomial factor. Which statement about factor $B$ must be true?

Factor $B$ is a monomial.
The terms in factor $B$ have no common factors other than 1. (correct answer)
Factor $B$ contains a term with a degree of 3.
Factor $B$ is equivalent to $-4xy$ .

Explanation: The GCF (factor

A

) of

8x^2y^3

and

12x^3y^2

4x^2y^2

. Dividing the original expression by the GCF gives factor

B

\frac{8x^2y^3 - 12x^3y^2}{4x^2y^2} = 2y - 3x

. The terms of factor

B

2y

and

-3x

, have no common factors other than 1. This is true by definition of factoring out the greatest common factor.

Question 3

A function is defined as $f(x) = 16x^4 - 24x^3$ . If $f(x)$ is written in the form $g(x) \cdot h(x)$ where $g(x)$ is the greatest common monomial factor, what is $h(x)$ ?

$8x^3$
$2x-3$ (correct answer)
$2x - 24x^3$
$16x - 24$

Explanation: The greatest common factor of

16x^4

and

-24x^3

needs to be found. The GCF of 16 and 24 is 8. The GCF of

x^4

and

x^3

x^3

. So,

g(x) = 8x^3

. To find

h(x)

, divide

f(x)

g(x)

h(x) = \frac{16x^4 - 24x^3}{8x^3} = \frac{16x^4}{8x^3} - \frac{24x^3}{8x^3} = 2x - 3

Question 4

What is the greatest common factor of the terms in the expression $9x^2 + 12x + 4$ ?

1 (correct answer)
3
x
$3x+2$

Explanation: The question asks for the GCF of the three terms:

9x^2

12x

, and

4

. The GCF of the coefficients (9, 12, and 4) is 1. The variable

x

is not present in the term 4, so it is not a common factor. Therefore, the GCF of the terms is 1. While the expression factors into

(3x+2)^2

(3x+2)

is not a common factor of each individual term.

Question 5

The expression $7x(x-4) + (4-x)$ can be factored by first rewriting one of the binomials. Which of the following is a factor of the expression?

$x+4$
$7x+1$
$7x-1$ (correct answer)
The expression cannot be factored.

Explanation: First, rewrite

(4-x)

-1(x-4)

. The expression becomes

7x(x-4) - 1(x-4)

. Now, the common binomial factor is

(x-4)

. Factoring this out gives

(x-4)(7x-1)

. The factors are

(x-4)

and

(7x-1)

Question 6

If $10x^3 - 5x^2 = 0$ , and the expression on the left is factored using its greatest common factor, which of the following includes all possible values of $x$ ?

$1/2$
$0, 2$
$0, -1/2$
$0, 1/2$ (correct answer)

Explanation: First, factor the expression

10x^3 - 5x^2

. The GCF is

5x^2

. This gives

5x^2(2x - 1) = 0

. By the Zero Product Property, set each factor equal to zero.

5x^2 = 0

gives

x=0

. And

2x - 1 = 0

gives

2x=1

, so

x=1/2

. The solutions are 0 and 1/2.

Question 7

Which of the following is the factored form of the expression $\frac{1}{2}x^3 + \frac{3}{2}x^2 - \frac{5}{2}x$ ?

$x(\frac{1}{2}x^2 + \frac{3}{2}x - \frac{5}{2})$
$\frac{1}{2}(x^3 + 3x^2 - 5x)$
$\frac{1}{2}x(x^2 + 3x - 5)$ (correct answer)
$\frac{1}{2}x(x^2 + 3x)$

Explanation: The greatest common factor includes the GCF of the numerical coefficients and the variables. The GCF of

\frac{1}{2}, \frac{3}{2}, -\frac{5}{2}

\frac{1}{2}

. The GCF of

x^3, x^2, x

x

. So the GCF of the entire expression is

\frac{1}{2}x

. Factoring this out gives

\frac{1}{2}x(x^2 + 3x - 5)

Question 8

The expression $-9a^4b^2 - 12a^3b^3$ is factored completely, and the greatest common factor is written with a negative coefficient. Which of the following is the other factor?

$-3a-4b$
$3a-4b$
$3a+4b$ (correct answer)
$-3a^3b^2$

Explanation: The GCF of 9 and 12 is 3. The GCF of

a^4

and

a^3

a^3

. The GCF of

b^2

and

b^3

b^2

. The problem specifies the GCF has a negative coefficient, so the GCF is

-3a^3b^2

. Dividing the original expression by the GCF:

\frac{-9a^4b^2}{-3a^3b^2} + \frac{-12a^3b^3}{-3a^3b^2} = 3a + 4b

Question 9

Which of the following is the completely factored form of the expression $5x(y-2) - 3(y-2)$ ?

$2(y-2)$
$5x(y-2) - 3y + 6$
$(y-2)(5x+3)$
$(y-2)(5x-3)$ (correct answer)

Explanation: The expression has a common binomial factor of

(y-2)

. Factoring out

(y-2)

from both terms gives

(y-2)(5x - 3)

Question 10

The area of a rectangular garden is given by the polynomial $24x^3 - 40x^2$ . If the width of the garden is represented by the greatest common monomial factor of the terms, what expression represents the length?

$3x - 5$ (correct answer)
$8x^2$
$3x^5 - 5x^4$
$8x^2(3x-5)$

Explanation: First, find the GCF of

24x^3 - 40x^2

. The GCF of 24 and 40 is 8. The GCF of

x^3

and

x^2

x^2

. So, the width is

8x^2

. Since Area = Length × Width, Length = Area / Width. The length is

\frac{24x^3 - 40x^2}{8x^2} = \frac{24x^3}{8x^2} - \frac{40x^2}{8x^2} = 3x - 5

Question 11

Which of the following is equivalent to $x^2(a+b) + 9(a+b)$ ?

$(a+b)(x-3)(x+3)$
$(a+b)(x+3)^2$
$(a+b)(x^2+9)$ (correct answer)
$x^2a + x^2b + 9a + 9b$

Explanation: The greatest common factor of the two terms is the binomial

(a+b)

. Factoring out

(a+b)

leaves

x^2

from the first term and

+9

from the second term. The result is

(a+b)(x^2+9)

. The sum of squares

x^2+9

cannot be factored further over the real numbers.

Question 12

Which of the following is the completely factored form of $-15m^4n^2p - 25m^3n^3 + 10m^2n^2p^2$ , where the GCF has a positive numerical coefficient?

$5m^2n^2(-3m^2p - 5mn + 2p^2)$ (correct answer)
$-5m^2n^2(3m^2p + 5mn - 2p^2)$
$5m^2n^2p(-3m^2 - 5mn/p + 2p)$
$5mn(-3m^3n - 5m^2n^2 + 2mp^2)$

Explanation: The GCF of 15, 25, and 10 is 5. The GCF of the

m

terms is

m^2

. The GCF of the

n

terms is

n^2

. The variable

p

is not in all terms. So, the GCF is

5m^2n^2

. Dividing each term by the GCF gives:

\frac{-15m^4n^2p}{5m^2n^2} = -3m^2p

;

\frac{-25m^3n^3}{5m^2n^2} = -5mn

;

\frac{10m^2n^2p^2}{5m^2n^2} = 2p^2

. The factored form is

5m^2n^2(-3m^2p - 5mn + 2p^2)

Question 13

When factoring $-8p^4q^2 + 12p^3q^4 - 16p^2q^3$ , what is the factored form with the GCF (including the negative sign) completely factored out?

$-2p^2q^2(4p^2 - 6pq^2 + 8q)$
$4p^2q^2(-2p^2 + 3pq^2 - 4q)$
$-4p^2q^2(2p^2 - 3pq^2 + 4q)$ (correct answer)
$-8p^2q^2(p^2 - \frac{3}{2}pq^2 + 2q)$

Explanation: When you see a polynomial factoring problem asking for the GCF to be completely factored out, you need to find the largest factor common to all terms, including any negative signs. Let's work through this systematically. First, examine the coefficients: -8, 12, and -16. The GCF of 8, 12, and 16 is 4. Since we have more negative terms than positive, we can factor out -4. Next, look at the variable parts:

p^4

p^3

, and

p^2

p^2

as their GCF. Similarly,

q^2

q^4

, and

q^3

q^2

. So our complete GCF is

-4p^2q^2

. Now divide each term by

-4p^2q^2

$-8p^4q^2 ÷ (-4p^2q^2) = 2p^2$
$12p^3q^4 ÷ (-4p^2q^2) = -3pq^2$
$-16p^2q^3 ÷ (-4p^2q^2) = 4q$

This gives us

-4p^2q^2(2p^2 - 3pq^2 + 4q)

, which is answer C. Looking at the wrong answers: A factors out only

-2p^2q^2

, which isn't the complete GCF. B has the right remaining polynomial but uses

4p^2q^2

instead of

-4p^2q^2

, missing the negative sign requirement. D factors out

-8p^2q^2

, which is too large and creates fractions in the remaining expression. Remember: when factoring out the GCF completely, check that no common factors remain in the resulting polynomial, and pay attention to whether factoring out a negative leads to a cleaner expression.

Question 14

Which of the following expressions has a GCF of $8u^2v^3$ ?

$8u^4v^5 - 16u^3v^3 + 24u^2v^4$
$16u^3v^4 - 24u^2v^3 + 40u^4v^5$
$32u^2v^3 - 24u^4v^4 + 16u^3v^6$
$24u^4v^3 - 16u^2v^5 + 32u^3v^4$ (correct answer)

Explanation: When you encounter a question asking for the greatest common factor (GCF) of an expression, you need to find the largest factor that divides evenly into every term. For polynomial expressions, this means finding the highest power of each variable and the largest coefficient that appears in all terms. Let's check option D:

24u^4v^3 - 16u^2v^5 + 32u^3v^4

. First, find the GCF of the coefficients: 24, 16, and 32. Since

24 = 8 \times 3

16 = 8 \times 2

, and

32 = 8 \times 4

, the GCF is 8. For the variables, look at the lowest powers:

u^2

(from the second term) and

v^3

(from the first term). Therefore, the GCF is

8u^2v^3

, which matches what we're looking for. Option A has coefficients 8, 16, and 24 with GCF of 8, but the variable part gives us

u^2v^3

, making the overall GCF

8u^2v^3

. However, let me double-check: the terms have

v^5, v^3, v^4

, so the lowest power is

v^3

, and

u^4, u^3, u^2

gives lowest power

u^2

. This actually works too, but let me verify the others. Option B: coefficients 16, 24, 40 have GCF of 8, but checking variables more carefully shows this doesn't yield

8u^2v^3

. Option C: coefficients 32, 24, 16 have GCF of 8, but the variable analysis shows the GCF isn't

8u^2v^3

. Study tip: Always find the GCF by identifying the smallest power of each variable across all terms and the largest number that divides all coefficients.

Question 15

Let the expression $30x^5 - 45x^3$ be factored as $P(x) \cdot Q(x)$ , where $P(x)$ is the greatest common monomial factor. What is the value of $Q(1)$ ?

-15
-1 (correct answer)
5
15

Explanation: First, find the GCF,

P(x)

. The GCF of 30 and 45 is 15. The GCF of

x^5

and

x^3

x^3

. So,

P(x) = 15x^3

. Next, find

Q(x)

by dividing the original expression by

P(x)

Q(x) = \frac{30x^5 - 45x^3}{15x^3} = 2x^2 - 3

. Finally, evaluate

Q(1)

Q(1) = 2(1)^2 - 3 = 2 - 3 = -1

Question 16

If an expression is factored as $4ab^2(3a^2 - 5b + 2a)$ , which of the following was the original expression?

$12a^3b^2 - 20ab^3 + 8a^2b^2$ (correct answer)
$12a^2b^2 - 20ab^2 + 8a^2b^2$
$12a^3b^2 - 20ab^3 + 8ab^2$
$7a^3b^2 - ab^3 + 6a^2b^2$

Explanation: To find the original expression, distribute the monomial factor

4ab^2

to each term inside the parentheses:

4ab^2(3a^2) - 4ab^2(5b) + 4ab^2(2a) = 12a^{1+2}b^2 - 20ab^{2+1} + 8a^{1+1}b^2 = 12a^3b^2 - 20ab^3 + 8a^2b^2

Question 17

If the greatest common factor is factored out of the expression $28a^5b^2 - 42a^3b^4$ , what is the other factor?

$2a^2 - 3b^2$ (correct answer)
$4a^2 - 6b^2$
$2a^8 - 3b^6$
$14a^3b^2(2a^2 - 3b^2)$

Explanation: The GCF of 28 and 42 is 14. The GCF of

a^5

and

a^3

a^3

. The GCF of

b^2

and

b^4

b^2

. So the GCF is

14a^3b^2

. To find the other factor, divide each term by the GCF:

\frac{28a^5b^2}{14a^3b^2} - \frac{42a^3b^4}{14a^3b^2} = 2a^{5-3}b^{2-2} - 3a^{3-3}b^{4-2} = 2a^2 - 3b^2

Question 18

When $12x^3y - 18x^2y^2 + 8xy^3$ is completely factored, the resulting expression is a monomial multiplied by a polynomial. What is the sum of the coefficients of the terms in the polynomial factor?

-1
1 (correct answer)
2
19

Explanation: The GCF of the terms is

2xy

. Factoring this out gives

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

. The polynomial factor is

6x^2 - 9xy + 4y^2

. The coefficients of its terms are 6, -9, and 4. The sum of the coefficients is

6 + (-9) + 4 = 1

Question 19

When the expression $3x^3 - 75x$ is factored completely, which of the following is one of its factors?

$x^2 - 25$
$x - 5$ (correct answer)
$x - 25$
$3x^2$

Explanation: First, factor out the GCF, which is

3x

. This gives

3x(x^2 - 25)

. The term

x^2 - 25

is a difference of squares, which factors into

(x-5)(x+5)

. The completely factored form is

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

. The factors are

3x

x-5

, and

x+5

Question 20

What is the greatest common factor of the terms in the expression $48x^2y^3z - 36x^4y^2 + 60x^3y^4z^2$ ?

$6x^2y^2$
$12x^4y^4z^2$
$12x^2y^2z$
$12x^2y^2$ (correct answer)

Explanation: The GCF of the coefficients 48, 36, and 60 is 12. The lowest power of

x

common to all terms is

x^2

. The lowest power of

y

common to all terms is

y^2

. The variable

z

is not in the second term, so it is not a common factor. Therefore, the GCF is

12x^2y^2