ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Distributing And Factoring Gcf
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Distributing And Factoring GcfQuestion 1 of 20

Which expression is equivalent to −15m3n+25m2n2−35mn3-15m^3n + 25m^2n^2 - 35mn^3 after factoring out the GCF?

5mn(3m2−5mn+7n2)5mn(3m^2 - 5mn + 7n^2)
5mn(−3m2+5mn−7n2)5mn(-3m^2 + 5mn - 7n^2)
−5mn(3m2+5mn+7n2)-5mn(3m^2 + 5mn + 7n^2)
−5mn(3m2−5mn+7n2)-5mn(3m^2 - 5mn + 7n^2)
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Distributing And Factoring Gcf

Practice Distributing And Factoring Gcf in ACCUPLACER Quantitative Reasoning, Algebra & Statistics 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 Distributing And Factoring Gcf, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

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

Which expression is equivalent to −15m3n+25m2n2−35mn3-15m^3n + 25m^2n^2 - 35mn^3 after factoring out the GCF?

  1. 5mn(3m2−5mn+7n2)5mn(3m^2 - 5mn + 7n^2)
  2. 5mn(−3m2+5mn−7n2)5mn(-3m^2 + 5mn - 7n^2)
  3. −5mn(3m2+5mn+7n2)-5mn(3m^2 + 5mn + 7n^2)
  4. −5mn(3m2−5mn+7n2)-5mn(3m^2 - 5mn + 7n^2) (correct answer)
Explanation: When you see a polynomial expression that needs factoring, your goal is to identify the Greatest Common Factor (GCF) among all terms and factor it out completely. To find the GCF of −15m3n+25m2n2−35mn3-15m^3n + 25m^2n^2 - 35mn^3, examine both the coefficients and variables separately. For coefficients: the GCF of 15, 25, and 35 is 5. For variables: each term contains at least m1m^1 and n1n^1, so the variable part of the GCF is mnmn. However, since the first term is negative, you can factor out −5mn-5mn to avoid having a negative leading coefficient in the parentheses. Factoring out −5mn-5mn:
  • −15m3n÷(−5mn)=3m2-15m^3n ÷ (-5mn) = 3m^2
  • 25m2n2÷(−5mn)=−5mn25m^2n^2 ÷ (-5mn) = -5mn
  • −35mn3÷(−5mn)=7n2-35mn^3 ÷ (-5mn) = 7n^2
This gives us −5mn(3m2−5mn+7n2)-5mn(3m^2 - 5mn + 7n^2), which is answer D. Let's check why the other options are incorrect: Choice A uses 5mn5mn instead of −5mn-5mn as the GCF, which would change the signs inside the parentheses incorrectly. Choice B has the right GCF but wrong signs throughout the parentheses—this happens when students make sign errors during division. Choice C incorrectly shows addition signs instead of the proper subtraction, suggesting confusion about how negative division affects the terms. Study tip: Always verify your factoring by distributing the GCF back through the parentheses. If you get the original expression, your factorization is correct.

Question 2

A triangle has side lengths of (4x+8)(4x+8), (6x+12)(6x+12), and (2x+4)(2x+4). Which of the following expressions represents the perimeter of the triangle in factored form?

  1. 12x+2412x+24
  2. 4(3x+6)4(3x+6)
  3. 6(2x+4)6(2x+4)
  4. 12(x+2)12(x+2) (correct answer)
Explanation: First, find the perimeter by adding the lengths of the three sides: (4x+8)+(6x+12)+(2x+4)(4x+8) + (6x+12) + (2x+4). Combine like terms: (4x+6x+2x)+(8+12+4)=12x+24(4x+6x+2x) + (8+12+4) = 12x + 24. Next, factor the expression for the perimeter. The greatest common factor of 12x12x and 2424 is 12. Factoring out 12 gives 12(x+2)12(x+2).

Question 3

Which of the following expressions is NOT equivalent to 18n3−6n2+12n18n^3 - 6n^2 + 12n?

  1. 6n(3n2−n+2)6n(3n^2 - n + 2)
  2. 3n(6n2−2n+4)3n(6n^2 - 2n + 4)
  3. 2(9n3−3n2+6n)2(9n^3 - 3n^2 + 6n)
  4. −6n(−3n2+n+2)-6n(-3n^2 + n + 2) (correct answer)
Explanation: Check each option by distributing. A: 6n(3n2−n+2)=18n3−6n2+12n6n(3n^2 - n + 2) = 18n^3 - 6n^2 + 12n (Equivalent). B: 3n(6n2−2n+4)=18n3−6n2+12n3n(6n^2 - 2n + 4) = 18n^3 - 6n^2 + 12n (Equivalent). C: 2(9n3−3n2+6n)=18n3−6n2+12n2(9n^3 - 3n^2 + 6n) = 18n^3 - 6n^2 + 12n (Equivalent). D: −6n(−3n2+n+2)=18n3−6n2−12n-6n(-3n^2 + n + 2) = 18n^3 - 6n^2 - 12n. The last term is −12n-12n instead of +12n+12n, so this expression is not equivalent.

Question 4

The area of a rectangular garden is given by the expression 5(3x−2)5(3x - 2) square meters. The width of the garden is 5 meters. If the expression for the area is expanded to 15x−1015x - 10, what does the term 15x15x represent in this context?

  1. The length of the garden
  2. The perimeter of the garden
  3. The area of a smaller rectangle that is removed from a larger piece
  4. The area of a larger rectangle with dimensions 5 meters by 3x3x meters (correct answer)
Explanation: The area of a rectangle is length times width. Here, the width is 5 and the length is (3x−2)(3x - 2). The expanded area, 15x−1015x - 10, can be interpreted as the area of a rectangle with width 5 and length 3x3x (which is 15x15x) from which a smaller rectangular area of 5×2=105 \times 2 = 10 has been subtracted. Therefore, the term 15x15x represents the area of that larger, initial rectangle.

Question 5

Which of the following expressions is equivalent to x(3x+4)−2(3x+4)x(3x + 4) - 2(3x + 4)?

  1. (x−2)(3x+4)(x-2)(3x+4) (correct answer)
  2. 3x2−2x−83x^2 - 2x - 8
  3. 3x2−83x^2 - 8
  4. 3x2+10x+83x^2 + 10x + 8
Explanation: This expression can be factored by recognizing that (3x+4)(3x+4) is a common factor. Factoring out (3x+4)(3x+4) gives (x−2)(3x+4)(x-2)(3x+4). To verify: expanding (x−2)(3x+4)(x-2)(3x+4) yields 3x2+4x−6x−8=3x2−2x−83x^2 + 4x - 6x - 8 = 3x^2 - 2x - 8, but the factored form (x−2)(3x+4)(x-2)(3x+4) is the most simplified equivalent expression.

Question 6

A large rectangular poster has a length of 5x5x inches and a width of 3x+23x+2 inches. A smaller rectangular photo with an area of 6x6x square inches is placed on top of the poster. Which expression represents the visible area of the poster?

  1. 15x2+10x15x^2 + 10x
  2. 15x2+16x15x^2 + 16x
  3. 15x2+4x15x^2 + 4x (correct answer)
  4. 2x+22x+2
Explanation: First, find the total area of the poster by multiplying its length and width: 5x(3x+2)5x(3x+2). Distributing gives 15x2+10x15x^2 + 10x. The visible area is the total area of the poster minus the area of the photo. So, subtract 6x6x from the total area: (15x2+10x)−6x(15x^2 + 10x) - 6x. Combining like terms (10x−6x=4x10x - 6x = 4x) gives the final expression 15x2+4x15x^2 + 4x.

Question 7

The profit, in dollars, from selling nn items is given by the expression 20n−(5n+50)20n - (5n + 50). Which of the following is an equivalent expression for the profit?

  1. 15n+5015n + 50
  2. 15n−5015n - 50 (correct answer)
  3. 25n+5025n + 50
  4. 20n−5n+5020n - 5n + 50
Explanation: To simplify the expression 20n−(5n+50)20n - (5n + 50), distribute the negative sign to both terms inside the parentheses. This changes the expression to 20n−5n−5020n - 5n - 50. Then, combine the like terms: 20n−5n=15n20n - 5n = 15n. The simplified expression is 15n−5015n - 50.

Question 8

The total cost for a group to rent a bus is 100+4n100 + 4n dollars, where nn is the number of people. The total revenue from the trip is 10n10n dollars. If profit is revenue minus cost, which expression represents the profit for the trip in factored form?

  1. 6n−1006n - 100
  2. 14n+10014n + 100
  3. 2(3n−50)2(3n - 50) (correct answer)
  4. 4(n−25)4(n - 25)
Explanation: Profit is calculated as Revenue - Cost. The expression for profit is 10n−(100+4n)10n - (100 + 4n). First, distribute the negative sign: 10n−100−4n10n - 100 - 4n. Then, combine like terms: 10n−4n=6n10n - 4n = 6n, so the profit is 6n−1006n - 100. To write this in factored form, find the greatest common factor of 6n6n and 100, which is 2. Factoring out 2 gives 2(3n−50)2(3n - 50).

Question 9

A student claims that 4x2−8x+124x^2 - 8x + 12 factors as 4(x2−2x+3)4(x^2 - 2x + 3). What error, if any, did the student make?

  1. No error; the factorization is completely correct (correct answer)
  2. The student should have factored out 2 instead of 4
  3. The student factored correctly but could factor further since x2−2x+3x^2 - 2x + 3 factors
  4. The student factored correctly and found the complete factorization
Explanation: The student correctly factored out the GCF of 4: 4x² - 8x + 12 = 4(x² - 2x + 3). Choice B is wrong because 4 is indeed the GCF of 4, 8, and 12. Choice C is incorrect because x² - 2x + 3 cannot factor further over the reals (discriminant = 4 - 12 = -8 < 0). Choice D is redundant with A but uses different wording that might confuse students about whether further factoring is possible.

Question 10

Two students factor the same expression differently: Student A gets 6x(2x2−3x+1)6x(2x^2 - 3x + 1) and Student B gets 3x(4x2−6x+2)3x(4x^2 - 6x + 2). If both factorizations are mathematically correct, what can be concluded?

  1. Student B factored out the GCF completely, but Student A did not
  2. Student A factored out the GCF completely, but Student B did not (correct answer)
  3. Both students factored out the GCF completely and both are fully correct
  4. Neither student factored correctly since they got different results
Explanation: When you encounter factoring problems where students get different results, you need to check whether each factorization is complete by examining the greatest common factor (GCF). Let's verify both factorizations by expanding them. Student A: 6x(2x2−3x+1)=12x3−18x2+6x6x(2x^2 - 3x + 1) = 12x^3 - 18x^2 + 6x. Student B: 3x(4x2−6x+2)=12x3−18x2+6x3x(4x^2 - 6x + 2) = 12x^3 - 18x^2 + 6x. Both give the same result, so both factorizations are mathematically correct. However, the key difference is completeness. Student A factored out 6x6x, leaving the expression (2x2−3x+1)(2x^2 - 3x + 1) inside the parentheses. Looking at the coefficients 2, -3, and 1, their GCF is 1, so no further factoring is possible. Student B factored out only 3x3x, leaving (4x2−6x+2)(4x^2 - 6x + 2). The coefficients 4, -6, and 2 have a GCF of 2, meaning Student B could factor out an additional 2: 3x(4x2−6x+2)=3x⋅2(2x2−3x+1)=6x(2x2−3x+1)3x(4x^2 - 6x + 2) = 3x \cdot 2(2x^2 - 3x + 1) = 6x(2x^2 - 3x + 1). Choice A is backwards—Student A found the complete GCF. Choice C is wrong because Student B didn't factor completely. Choice D incorrectly assumes different-looking results can't both be mathematically valid. Choice B correctly identifies that Student A factored out the complete GCF while Student B stopped partway. Remember: when factoring, always check if the terms inside parentheses share any common factors. Complete factorization means no further common factors remain.

Question 11

The surface area of a cylinder is given by the formula A=2πr2+2πrhA = 2\pi r^2 + 2\pi rh. Which of the following shows the formula with the greatest common factor factored out?

  1. 2π(r2+rh)2\pi(r^2 + rh)
  2. 2πr(r+h)2\pi r(r + h) (correct answer)
  3. 2r(πr+πh)2r(\pi r + \pi h)
  4. 2πr2(1+h)2\pi r^2(1 + h)
Explanation: In the expression 2πr2+2πrh2\pi r^2 + 2\pi rh, we look for the greatest common factor (GCF) of the two terms. Both terms share the factors 2, π\pi, and rr. Therefore, the GCF is 2πr2\pi r. Factoring 2πr2\pi r out of the first term, 2πr22\pi r^2, leaves rr. Factoring it out of the second term, 2πrh2\pi rh, leaves hh. The factored form is 2πr(r+h)2\pi r(r + h).

Question 12

Which of the following expressions is equivalent to 7x−3(4x−2)7x - 3(4x - 2)?

  1. −5x−6-5x - 6
  2. −5x−2-5x - 2
  3. −5x+6-5x + 6 (correct answer)
  4. 19x−619x - 6
Explanation: To simplify the expression, first distribute the -3 to both terms inside the parentheses: −3×4x=−12x-3 \times 4x = -12x and −3×−2=+6-3 \times -2 = +6. The expression becomes 7x−12x+67x - 12x + 6. Then, combine the like terms: 7x−12x=−5x7x - 12x = -5x. The simplified expression is −5x+6-5x + 6.

Question 13

Which of the following expressions is equivalent to 12x4y2−18x2y312x^4y^2 - 18x^2y^3?

  1. 3x2y2(4x2−6y)3x^2y^2(4x^2 - 6y)
  2. 6x2y(2x2y−3y2)6x^2y(2x^2y - 3y^2)
  3. 6x2y2(2x2−3y)6x^2y^2(2x^2 - 3y) (correct answer)
  4. 6xy(2x3y−3xy2)6xy(2x^3y - 3xy^2)
Explanation: To factor the expression completely, find the greatest common factor (GCF) of both terms. The GCF of the coefficients 12 and 18 is 6. The GCF of the variable parts x4y2x^4y^2 and x2y3x^2y^3 is x2y2x^2y^2 (the lowest power of each variable). So, the GCF of the entire expression is 6x2y26x^2y^2. Factoring this out gives 6x2y2(2x2−3y)6x^2y^2(2x^2 - 3y).

Question 14

Which of the following is equivalent to −8a3b−12ab2-8a^3b - 12ab^2?

  1. −2ab(4a2+6b)-2ab(4a^2 + 6b)
  2. −4ab(2a2−3b)-4ab(2a^2 - 3b)
  3. −4ab(2a2+3b)-4ab(2a^2 + 3b) (correct answer)
  4. 4ab(−2a2+3b)4ab(-2a^2 + 3b)
Explanation: The greatest common factor (GCF) of 8 and 12 is 4. The GCF of a3a^3 and aa is aa. The GCF of bb and b2b^2 is bb. So the GCF is 4ab4ab. Since the leading term is negative, we factor out −4ab-4ab. Dividing each term by −4ab-4ab: −8a3b−4ab=2a2\frac{-8a^3b}{-4ab} = 2a^2 and −12ab2−4ab=3b\frac{-12ab^2}{-4ab} = 3b. This gives the factored form −4ab(2a2+3b)-4ab(2a^2 + 3b).

Question 15

An expression is given by 7−(4x−5)7 - (4x - 5). Which of the following is an equivalent expression?

  1. 2−4x2 - 4x
  2. 12−4x12 - 4x (correct answer)
  3. 7−4x−57 - 4x - 5
  4. 3x−53x - 5
Explanation: The expression 7−(4x−5)7 - (4x - 5) is equivalent to 7−1(4x−5)7 - 1(4x - 5). Distribute the -1 to both terms inside the parentheses: −1×4x=−4x-1 \times 4x = -4x and −1×−5=+5-1 \times -5 = +5. The expression becomes 7−4x+57 - 4x + 5. Combine the constant terms 7+5=127 + 5 = 12 to get the final simplified expression 12−4x12 - 4x.

Question 16

The expression 24x3−36x224x^3 - 36x^2 can be written in the form A⋅BA \cdot B, where AA is the greatest common monomial factor. What is the expression for BB?

  1. 12x212x^2
  2. 2x−32x - 3 (correct answer)
  3. 2x3−3x22x^3 - 3x^2
  4. 12x−1812x - 18
Explanation: First, find the greatest common factor (GCF) of 24x324x^3 and 36x236x^2. The GCF of 24 and 36 is 12. The GCF of x3x^3 and x2x^2 is x2x^2. Thus, the GCF, which is AA, is 12x212x^2. To find BB, divide the original expression by AA: 24x3−36x212x2=24x312x2−36x212x2=2x−3\frac{24x^3 - 36x^2}{12x^2} = \frac{24x^3}{12x^2} - \frac{36x^2}{12x^2} = 2x - 3. So, B=2x−3B = 2x - 3.

Question 17

Which of the following is equivalent to 23(6x−9y+12)\frac{2}{3}(6x - 9y + 12)?

  1. 4x−6y+84x - 6y + 8 (correct answer)
  2. 4x−9y+124x - 9y + 12
  3. 4x+6y+84x + 6y + 8
  4. 9x−272y+189x - \frac{27}{2}y + 18
Explanation: To find the equivalent expression, distribute 23\frac{2}{3} to each term inside the parentheses. 23×6x=123x=4x\frac{2}{3} \times 6x = \frac{12}{3}x = 4x. 23×(−9y)=−183y=−6y\frac{2}{3} \times (-9y) = -\frac{18}{3}y = -6y. 23×12=243=8\frac{2}{3} \times 12 = \frac{24}{3} = 8. Combining these terms gives 4x−6y+84x - 6y + 8.

Question 18

An expression is given by 3x(y−2)+5(y−2)3x(y-2) + 5(y-2). If this expression is factored completely, which of the following is one of its factors?

  1. y+2y+2
  2. 3x−53x-5
  3. 3x+53x+5 (correct answer)
  4. 3xy−6x+53xy-6x+5
Explanation: This expression can be factored by grouping. Notice that (y−2)(y-2) is a common binomial factor to both terms. Factoring out (y−2)(y-2) leaves 3x3x from the first term and +5+5 from the second term. The completely factored expression is (y−2)(3x+5)(y-2)(3x+5). The two factors are (y−2)(y-2) and (3x+5)(3x+5).

Question 19

Which of the following is the result of distributing −3x2y-3x^2y to the expression (5xy2−2x2)(5xy^2 - 2x^2)?

  1. −15x3y3−6x4y-15x^3y^3 - 6x^4y
  2. −15x3y3+6x4y-15x^3y^3 + 6x^4y (correct answer)
  3. −15x2y2+6x4y-15x^2y^2 + 6x^4y
  4. 2x3y3−5x4y2x^3y^3 - 5x^4y
Explanation: Apply the distributive property. Multiply −3x2y-3x^2y by each term in the parentheses. For the first term: (−3x2y)(5xy2)=(−3)(5)(x2⋅x)(y⋅y2)=−15x3y3(-3x^2y)(5xy^2) = (-3)(5)(x^2 \cdot x)(y \cdot y^2) = -15x^3y^3. For the second term: (−3x2y)(−2x2)=(−3)(−2)(x2⋅x2)(y)=6x4y(-3x^2y)(-2x^2) = (-3)(-2)(x^2 \cdot x^2)(y) = 6x^4y. The resulting expression is −15x3y3+6x4y-15x^3y^3 + 6x^4y.

Question 20

Which of the following expressions represents the phrase 'the product of 4x4x and the quantity 2x2x less than 7'?

  1. 28x−8x228x - 8x^2 (correct answer)
  2. 28x+8x228x + 8x^2
  3. 8x2−28x8x^2 - 28x
  4. 4x+7−2x4x + 7 - 2x
Explanation: First, translate the phrase into an algebraic expression. 'The quantity 2x2x less than 7' translates to 7−2x7 - 2x. 'The product of 4x4x and that quantity' means we multiply 4x4x by (7−2x)(7 - 2x), which is written as 4x(7−2x)4x(7 - 2x). To simplify, distribute the 4x4x: 4x×7=28x4x \times 7 = 28x and 4x×(−2x)=−8x24x \times (-2x) = -8x^2. The resulting expression is 28x−8x228x - 8x^2.