ACCUPLACER Advanced Algebra & Functions Quiz: Factoring Special Products
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Factoring Special ProductsQuestion 1 of 20

If the trinomial 9x2−30xy+25y29x^2 - 30xy + 25y^2 is factored as a perfect square, what is the result?

(3x−5y)2(3x - 5y)^2
(3x+5y)2(3x + 5y)^2
(3x−5y)(3x+5y)(3x - 5y)(3x + 5y)
(9x−5y)(x−5y)(9x - 5y)(x - 5y)
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ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Factoring Special Products

Practice Factoring Special Products 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 Special Products, 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

If the trinomial 9x2−30xy+25y29x^2 - 30xy + 25y^2 is factored as a perfect square, what is the result?

  1. (3x−5y)2(3x - 5y)^2 (correct answer)
  2. (3x+5y)2(3x + 5y)^2
  3. (3x−5y)(3x+5y)(3x - 5y)(3x + 5y)
  4. (9x−5y)(x−5y)(9x - 5y)(x - 5y)
Explanation: The expression 9x2−30xy+25y29x^2 - 30xy + 25y^2 is a perfect square trinomial of the form a2−2ab+b2a^2 - 2ab + b^2. Here, a2=9x2a^2 = 9x^2, so a=3xa = 3x, and b2=25y2b^2 = 25y^2, so b=5yb = 5y. The middle term is −2ab=−2(3x)(5y)=−30xy-2ab = -2(3x)(5y) = -30xy, which matches the expression. Therefore, it factors to (a−b)2=(3x−5y)2(a - b)^2 = (3x - 5y)^2.

Question 2

Which of the following is the factorization of 14x2−916\frac{1}{4}x^2 - \frac{9}{16}?

  1. (12x−34)(12x+34)(\frac{1}{2}x - \frac{3}{4})(\frac{1}{2}x + \frac{3}{4}) (correct answer)
  2. (14x−916)(14x+916)(\frac{1}{4}x - \frac{9}{16})(\frac{1}{4}x + \frac{9}{16})
  3. (12x−916)(12x+916)(\frac{1}{2}x - \frac{9}{16})(\frac{1}{2}x + \frac{9}{16})
  4. (12x−34)2(\frac{1}{2}x - \frac{3}{4})^2
Explanation: The expression 14x2−916\frac{1}{4}x^2 - \frac{9}{16} is a difference of two squares, a2−b2a^2 - b^2. The first term is a2=14x2a^2 = \frac{1}{4}x^2, so a=14x2=12xa = \sqrt{\frac{1}{4}x^2} = \frac{1}{2}x. The second term is b2=916b^2 = \frac{9}{16}, so b=916=34b = \sqrt{\frac{9}{16}} = \frac{3}{4}. The expression factors as (a−b)(a+b)(a-b)(a+b), which is (12x−34)(12x+34)(\frac{1}{2}x - \frac{3}{4})(\frac{1}{2}x + \frac{3}{4}).

Question 3

The area of a region is given by the expression 49x2−8149x^2 - 81. This area was formed by starting with a large square and removing a smaller square from it. Which expression could represent the perimeter of the original, larger square?

  1. 7x7x
  2. (7x−9)(7x+9)(7x-9)(7x+9)
  3. 3636
  4. 28x28x (correct answer)
Explanation: The area of the region is the difference between the area of the larger square and the area of the smaller square. The expression 49x2−8149x^2 - 81 represents this difference of areas. The area of the larger square is 49x249x^2, and the area of the smaller square is 8181. The side length of a square is the square root of its area. So, the side length of the larger square is 49x2=7x\sqrt{49x^2} = 7x. The perimeter of a square is 4 times its side length. Therefore, the perimeter of the larger square is 4(7x)=28x4(7x) = 28x.

Question 4

If p2−q2=48p^2 - q^2 = 48 and p+q=8p + q = 8, what is the value of p−qp - q?

  1. 6 (correct answer)
  2. 8
  3. 12
  4. 16
Explanation: Using the difference of squares factorization, p2−q2=(p+q)(p−q)p^2 - q^2 = (p + q)(p - q). Substituting the given values: 48=8(p−q)48 = 8(p - q). Solving for p−qp - q: p−q=488=6p - q = \frac{48}{8} = 6. Choice B incorrectly uses the value of p+qp + q. Choice C results from incorrectly dividing 48 by 4 instead of 8. Choice D results from incorrectly multiplying instead of dividing.

Question 5

The area of a rectangular garden is given by x2−14x+49x^2 - 14x + 49 square meters. If this expression represents a perfect square trinomial, what could be the length of one side of a square with the same area?

  1. 2x−72x - 7 meters
  2. x+7x + 7 meters
  3. x−14x - 14 meters
  4. x−7x - 7 meters (correct answer)
Explanation: When you encounter a perfect square trinomial, you're looking at an expression that can be factored into the form (a+b)2(a + b)^2 or (a−b)2(a - b)^2. The key is recognizing the pattern: the first and last terms are perfect squares, and the middle term equals twice the product of the square roots of those outer terms. Let's factor x2−14x+49x^2 - 14x + 49. The first term x2x^2 has a square root of xx, and the last term 4949 has a square root of 77. For a perfect square trinomial, the middle term should be 2⋅x⋅7=14x2 \cdot x \cdot 7 = 14x. Since we have −14x-14x, this factors as (x−7)2(x - 7)^2. You can verify: (x−7)2=x2−7x−7x+49=x2−14x+49(x - 7)^2 = x^2 - 7x - 7x + 49 = x^2 - 14x + 49 ✓ Since the garden's area is (x−7)2(x - 7)^2 square meters, a square with the same area would have side length x−7x - 7 meters, making D correct. Looking at the wrong answers: A) 2x−72x - 7 would give area (2x−7)2=4x2−28x+49(2x - 7)^2 = 4x^2 - 28x + 49, which doesn't match. B) x+7x + 7 would give area (x+7)2=x2+14x+49(x + 7)^2 = x^2 + 14x + 49—note the positive middle term. C) x−14x - 14 would give area (x−14)2=x2−28x+196(x - 14)^2 = x^2 - 28x + 196, completely different. Remember: when factoring perfect square trinomials, look for the pattern a2±2ab+b2=(a±b)2a^2 \pm 2ab + b^2 = (a \pm b)^2. The middle term's sign tells you whether to add or subtract in the factored form.

Question 6

Which of the following is equivalent to the expression x2−10x+25x2−25\frac{x^2 - 10x + 25}{x^2 - 25} for x≠5x \neq 5 and x≠−5x \neq -5?

  1. x+5x−5\frac{x+5}{x-5}
  2. x−5x+5\frac{x-5}{x+5} (correct answer)
  3. 1−25x1 - \frac{2}{5}x
  4. x−1x−1\frac{x-1}{x-1}
Explanation: To simplify the rational expression, factor both the numerator and the denominator. The numerator, x2−10x+25x^2 - 10x + 25, is a perfect square trinomial that factors to (x−5)2(x - 5)^2. The denominator, x2−25x^2 - 25, is a difference of squares that factors to (x−5)(x+5)(x - 5)(x + 5). The expression becomes (x−5)(x−5)(x−5)(x+5)\frac{(x - 5)(x - 5)}{(x - 5)(x + 5)}. Canceling the common factor of (x−5)(x - 5) leaves x−5x+5\frac{x-5}{x+5}.

Question 7

The expression 25x2+kx+1625x^2 + kx + 16 is a perfect square trinomial. Which of the following is a possible value for kk?

  1. 20
  2. 40 (correct answer)
  3. 80
  4. 100
Explanation: For a trinomial a2+2ab+b2a^2 + 2ab + b^2 to be a perfect square, its terms must fit the pattern. In 25x2+kx+1625x^2 + kx + 16, we have a2=25x2a^2 = 25x^2, so a=5xa = 5x, and b2=16b^2 = 16, so b=4b = 4. The middle term, kxkx, must be equal to 2ab2ab or −2ab-2ab. Calculating 2ab=2(5x)(4)=40x2ab = 2(5x)(4) = 40x. Therefore, a possible value for kk is 40. The other possible value would be -40.

Question 8

What is the complete factorization of (x2+4)2−(4x)2(x^2 + 4)^2 - (4x)^2?

  1. (x2−4x+4)(x2+4x+4)(x^2 - 4x + 4)(x^2 + 4x + 4)
  2. (x−2)4(x-2)^4
  3. (x−2)2(x+2)2(x-2)^2(x+2)^2 (correct answer)
  4. (x2−4)2(x^2 - 4)^2
Explanation: The expression is a difference of squares A2−B2A^2 - B^2 where A=x2+4A = x^2 + 4 and B=4xB = 4x. It factors into (A−B)(A+B)(A - B)(A + B), which is (x2+4−4x)(x2+4+4x)(x^2 + 4 - 4x)(x^2 + 4 + 4x). Rearranging the terms within the parentheses gives (x2−4x+4)(x2+4x+4)(x^2 - 4x + 4)(x^2 + 4x + 4). Both of these factors are perfect square trinomials. The first, x2−4x+4x^2 - 4x + 4, factors to (x−2)2(x-2)^2. The second, x2+4x+4x^2 + 4x + 4, factors to (x+2)2(x+2)^2. Thus, the complete factorization is (x−2)2(x+2)2(x-2)^2(x+2)^2.

Question 9

Which of the following is a factorization of −x2+6x−9-x^2 + 6x - 9?

  1. (x−3)2(x-3)^2
  2. −(x+3)2-(x+3)^2
  3. (−x−3)(x−3)(-x-3)(x-3)
  4. −(x−3)2-(x-3)^2 (correct answer)
Explanation: To factor −x2+6x−9-x^2 + 6x - 9, first factor out a -1 to make the leading coefficient positive: −(x2−6x+9)-(x^2 - 6x + 9). The expression in the parentheses, x2−6x+9x^2 - 6x + 9, is a perfect square trinomial a2−2ab+b2a^2 - 2ab + b^2, where a=xa=x and b=3b=3. This factors to (x−3)2(x-3)^2. Therefore, the complete factorization is −(x−3)2-(x-3)^2.

Question 10

If x2−y2=77x^2 - y^2 = 77 and x+y=11x + y = 11, what is the value of x−yx - y?

  1. -7
  2. 7 (correct answer)
  3. 11
  4. 66
Explanation: The expression x2−y2x^2 - y^2 is a difference of squares, which factors into (x−y)(x+y)(x - y)(x + y). We are given that x2−y2=77x^2 - y^2 = 77 and x+y=11x + y = 11. Substituting these values into the factored equation gives 77=(x−y)(11)77 = (x - y)(11). To find the value of x−yx - y, divide both sides of the equation by 11: 7711=x−y\frac{77}{11} = x - y, which simplifies to 7=x−y7 = x - y.

Question 11

If a−b=5a - b = 5 and a2−b2=65a^2 - b^2 = 65, what is the value of a+ba + b?

  1. 5
  2. 13 (correct answer)
  3. 60
  4. 325
Explanation: The formula for the difference of squares is a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). We are given the values for a2−b2a^2 - b^2 and a−ba - b. Substituting these into the formula gives 65=5(a+b)65 = 5(a + b). To solve for a+ba + b, divide both sides by 5: 655=a+b\frac{65}{5} = a + b. This gives 13=a+b13 = a + b.

Question 12

What is the solution to the equation 9x2+6x+1=09x^2 + 6x + 1 = 0?

  1. x=−3x = -3
  2. x=−13,x=13x = -\frac{1}{3}, x = \frac{1}{3}
  3. x=13x = \frac{1}{3}
  4. x=−13x = -\frac{1}{3} (correct answer)
Explanation: The expression on the left side of the equation, 9x2+6x+19x^2 + 6x + 1, is a perfect square trinomial. It fits the form a2+2ab+b2a^2 + 2ab + b^2 where a=3xa=3x and b=1b=1. It factors to (3x+1)2(3x + 1)^2. So the equation becomes (3x+1)2=0(3x + 1)^2 = 0. Taking the square root of both sides gives 3x+1=03x + 1 = 0. Subtracting 1 from both sides gives 3x=−13x = -1, and dividing by 3 gives the single solution x=−13x = -\frac{1}{3}.

Question 13

Which of the following expressions is NOT a perfect square trinomial or a difference of squares?

  1. x2−100x^2 - 100
  2. 4x2−12x+94x^2 - 12x + 9
  3. x2+8x+16x^2 + 8x + 16
  4. x2+25x^2 + 25 (correct answer)
Explanation: Let's analyze each choice. A) x2−100x^2 - 100 is a difference of squares: x2−102=(x−10)(x+10)x^2 - 10^2 = (x-10)(x+10). C) x2+8x+16x^2 + 8x + 16 is a perfect square trinomial: x2+2(x)(4)+42=(x+4)2x^2 + 2(x)(4) + 4^2 = (x+4)^2. D) 4x2−12x+94x^2 - 12x + 9 is a perfect square trinomial: (2x)2−2(2x)(3)+32=(2x−3)2(2x)^2 - 2(2x)(3) + 3^2 = (2x-3)^2. B) x2+25x^2 + 25 is a sum of squares, which cannot be factored into real linear factors. It is not a difference of squares or a perfect square trinomial.

Question 14

Which of the following is a complete factorization of the expression 12x2−7512x^2 - 75?

  1. 3(4x2−25)3(4x^2 - 25)
  2. 3(2x−5)(2x+5)3(2x - 5)(2x + 5) (correct answer)
  3. (6x−25)(2x+3)(6x - 25)(2x + 3)
  4. (2x−5)(2x+5)(2x - 5)(2x + 5)
Explanation: To factor the expression completely, first identify the greatest common factor (GCF) of the terms 12x212x^2 and −75-75, which is 3. Factoring out the GCF gives 3(4x2−25)3(4x^2 - 25). The expression inside the parentheses, 4x2−254x^2 - 25, is a difference of squares, a2−b2a^2 - b^2, where a=2xa = 2x and b=5b = 5. The difference of squares factors as (a−b)(a+b)(a - b)(a + b), so 4x2−25=(2x−5)(2x+5)4x^2 - 25 = (2x - 5)(2x + 5). Therefore, the complete factorization is 3(2x−5)(2x+5)3(2x - 5)(2x + 5).

Question 15

The expression 2x2+20x+502x^2 + 20x + 50 is a perfect square trinomial multiplied by a constant. What is its factored form?

  1. (2x+10)(x+5)(2x + 10)(x + 5)
  2. 2(x+5)22(x+5)^2 (correct answer)
  3. (x+5)2(x+5)^2
  4. 2(x+25)22(x+25)^2
Explanation: First, factor out the greatest common factor (GCF), which is 2: 2(x2+10x+25)2(x^2 + 10x + 25). The trinomial inside the parentheses, x2+10x+25x^2 + 10x + 25, is a perfect square trinomial of the form a2+2ab+b2a^2 + 2ab + b^2, where a=xa = x and b=5b = 5. The middle term 10x10x is equal to 2ab=2(x)(5)2ab = 2(x)(5). This trinomial factors to (a+b)2=(x+5)2(a + b)^2 = (x + 5)^2. Including the GCF, the complete factorization is 2(x+5)22(x+5)^2.

Question 16

What are all possible values of xx for which 4x2−49=04x^2 - 49 = 0?

  1. x=72x = \frac{7}{2}
  2. x=−72,x=72x = -\frac{7}{2}, x = \frac{7}{2} (correct answer)
  3. x=−27,x=27x = -\frac{2}{7}, x = \frac{2}{7}
  4. x=−494,x=494x = -\frac{49}{4}, x = \frac{49}{4}
Explanation: The equation 4x2−49=04x^2 - 49 = 0 can be solved by factoring the left side as a difference of squares, (2x)2−72=0(2x)^2 - 7^2 = 0, which gives (2x−7)(2x+7)=0(2x - 7)(2x + 7) = 0. By the zero product property, either 2x−7=02x - 7 = 0 or 2x+7=02x + 7 = 0. Solving the first equation gives 2x=72x = 7, or x=72x = \frac{7}{2}. Solving the second equation gives 2x=−72x = -7, or x=−72x = -\frac{7}{2}. Thus, there are two possible values for xx.

Question 17

For what value of cc is the expression x2+14x+cx^2 + 14x + c a perfect square trinomial?

  1. 7
  2. 28
  3. 49 (correct answer)
  4. 196
Explanation: A perfect square trinomial has the form a2+2ab+b2a^2 + 2ab + b^2. In the expression x2+14x+cx^2 + 14x + c, we can identify a2=x2a^2 = x^2, so a=xa = x. The middle term, 14x14x, must equal 2ab2ab. Substituting a=xa=x, we get 14x=2xb14x = 2xb. Dividing by 2x2x gives b=7b = 7. The constant term cc must be equal to b2b^2. Therefore, c=72=49c = 7^2 = 49.

Question 18

Which of the following is the complete factorization of 16x4−y416x^4 - y^4?

  1. (4x2−y2)(4x2+y2)(4x^2 - y^2)(4x^2 + y^2)
  2. (2x−y)(2x+y)(2x−y)(2x+y)(2x-y)(2x+y)(2x-y)(2x+y)
  3. (2x−y)(2x+y)(4x2+y2)(2x - y)(2x + y)(4x^2 + y^2) (correct answer)
  4. (4x−y)(4x+y)(x2+y2)(4x-y)(4x+y)(x^2+y^2)
Explanation: The expression 16x4−y416x^4 - y^4 is a difference of squares: (4x2)2−(y2)2(4x^2)^2 - (y^2)^2. This factors to (4x2−y2)(4x2+y2)(4x^2 - y^2)(4x^2 + y^2). The first factor, 4x2−y24x^2 - y^2, is also a difference of squares: (2x)2−y2(2x)^2 - y^2, which factors to (2x−y)(2x+y)(2x - y)(2x + y). The second factor, 4x2+y24x^2 + y^2, is a sum of squares and cannot be factored further over the real numbers. Combining these gives the complete factorization: (2x−y)(2x+y)(4x2+y2)(2x - y)(2x + y)(4x^2 + y^2).

Question 19

Which of the following is the complete factorization of 3x4−483x^4 - 48?

  1. 3(x2−4)(x2+4)3(x^2 - 4)(x^2 + 4)
  2. 3(x−2)(x+2)(x2+4)3(x - 2)(x + 2)(x^2 + 4) (correct answer)
  3. (3x2−12)(x2+4)(3x^2 - 12)(x^2 + 4)
  4. 3(x−2)2(x+2)23(x-2)^2(x+2)^2
Explanation: First, factor out the greatest common factor (GCF), which is 3: 3(x4−16)3(x^4 - 16). The expression inside the parentheses, x4−16x^4 - 16, is a difference of squares, (x2)2−42(x^2)^2 - 4^2, which factors to (x2−4)(x2+4)(x^2 - 4)(x^2 + 4). The factor x2−4x^2 - 4 is also a difference of squares, x2−22x^2 - 2^2, which factors to (x−2)(x+2)(x - 2)(x + 2). The factor x2+4x^2 + 4 is a sum of squares and cannot be factored further over real numbers. Therefore, the complete factorization is 3(x−2)(x+2)(x2+4)3(x - 2)(x + 2)(x^2 + 4).

Question 20

Which of the following is a factorization of (x+2)2−9(x+2)^2 - 9?

  1. (x−1)(x+5)(x - 1)(x + 5) (correct answer)
  2. (x−7)(x+11)(x - 7)(x + 11)
  3. x2+4x−5x^2 + 4x - 5
  4. (x−1)2(x - 1)^2
Explanation: The expression (x+2)2−9(x+2)^2 - 9 is a difference of squares, a2−b2a^2 - b^2, where a=(x+2)a = (x+2) and b=3b = 3. The factorization is (a−b)(a+b)(a - b)(a + b). Substituting the expressions for aa and bb gives ((x+2)−3)((x+2)+3)((x+2) - 3)((x+2) + 3). Simplifying inside each parenthesis results in (x−1)(x+5)(x - 1)(x + 5).