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Quiz: Factoring Polynomials

Test your understanding of factoring techniques and solving equations by factoring with these questions.

1. What is the greatest common factor (GCF) of the polynomial 12x³ + 18x²?

  1. 3x
  2. 6x
  3. 6x²
  4. 12x³
Show Answer

The correct answer is C. To find the GCF, identify the GCF of the coefficients (GCF(12,18) = 6) and the lowest power of x that appears in all terms (x²). Therefore, the GCF is 6x². Option A only includes part of the numerical GCF. Option B has the correct numerical part but the wrong variable power. Option D is one of the original terms, not the GCF.

Concept Tested: Factoring by GCF

See: Finding the GCF

2. Factor the trinomial x² + 7x + 12.

  1. (x + 2)(x + 6)
  2. (x + 1)(x + 12)
  3. (x + 3)(x + 4)
  4. (x + 7)(x + 5)
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The correct answer is C. We need two numbers that multiply to 12 and add to 7. The numbers 3 and 4 satisfy both conditions: 3 × 4 = 12 and 3 + 4 = 7, giving us (x + 3)(x + 4). Option A gives numbers that add to 8, not 7. Option B gives numbers that add to 13. Option D contains numbers that don't multiply to 12.

Concept Tested: Factoring Trinomials

See: Factoring Trinomials

3. Which expression demonstrates the difference of squares pattern?

  1. x² + 25
  2. x² - 10x + 25
  3. 9x² - 16
  4. x² + 6x + 9
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The correct answer is C. The difference of squares has the form a² - b². The expression 9x² - 16 = (3x)² - 4², which factors as (3x + 4)(3x - 4). Option A is a sum of squares (cannot be factored using real numbers). Options B and D are perfect square trinomials, not difference of squares.

Concept Tested: Factoring Difference of Squares

See: Difference of Squares

4. Factor x² - 25 completely.

  1. (x - 5)²
  2. (x + 5)²
  3. (x - 5)(x + 5)
  4. (x + 5)(x - 5)
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The correct answer is D. This is a difference of squares: x² - 25 = x² - 5² = (x + 5)(x - 5). Note that options C and D are mathematically equivalent due to the commutative property, but D follows the standard form (a + b)(a - b). Options A and B are perfect squares that would expand to trinomials, not the original binomial.

Concept Tested: Factoring Difference of Squares

See: Difference of Squares

5. Which trinomial is a perfect square?

  1. x² + 10x + 25
  2. x² + 7x + 12
  3. x² - 5x - 14
  4. x² + 3x + 9
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The correct answer is A. A perfect square trinomial has the form a² ± 2ab + b². For x² + 10x + 25: first term x² and last term 25 = 5² are perfect squares, and the middle term 10x = 2(x)(5) = 2ab. This factors as (x + 5)². Options B and C are factorable trinomials but not perfect squares. Option D has perfect square first and last terms, but 3x ≠ 2(x)(3) = 6x.

Concept Tested: Factoring Perfect Squares

See: Perfect Square Trinomials

6. Factor ax + ay + bx + by by grouping.

  1. (a + b)(x - y)
  2. (a - b)(x + y)
  3. a(x + y) + b(x + y)
  4. (a + b)(x + y)
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The correct answer is D. Group the terms: (ax + ay) + (bx + by). Factor each group: a(x + y) + b(x + y). Now factor out the common binomial (x + y): (a + b)(x + y). Option C shows the intermediate step but isn't fully factored. Options A and B have incorrect signs in the binomial factors.

Concept Tested: Factoring by Grouping

See: Factoring by Grouping

7. Solve the equation x² + 5x + 6 = 0 by factoring.

  1. x = 2 or x = 3
  2. x = -2 or x = -3
  3. x = 1 or x = 6
  4. x = -1 or x = -6
Show Answer

The correct answer is B. Factor the trinomial: (x + 2)(x + 3) = 0. Using the zero product property, set each factor equal to zero: x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3. Option A has the wrong signs (positive instead of negative). Options C and D use incorrect factor pairs for the constant 6.

Concept Tested: Solving by Factoring

See: Solving Equations by Factoring

8. What does the zero product property state?

  1. If a + b = 0, then a = 0 or b = 0
  2. If a - b = 0, then a = b
  3. If a × b = 0, then a = 0 or b = 0
  4. If a ÷ b = 0, then a = 0
Show Answer

The correct answer is C. The zero product property states that if the product of two or more factors equals zero, then at least one of the factors must equal zero: if ab = 0, then a = 0 or b = 0 (or both). This fundamental property allows us to solve factored equations. Options A, B, and D describe other algebraic properties but not the zero product property.

Concept Tested: Zero Product Property

See: The Zero Product Property

9. Factor 2x³ - 50x completely.

  1. 2x(x² - 25)
  2. 2x(x - 5)(x + 5)
  3. (2x - 10)(x + 5)
  4. 2(x³ - 25x)
Show Answer

The correct answer is B. First factor out the GCF of 2x: 2x(x² - 25). Then recognize that x² - 25 is a difference of squares that factors further: 2x(x + 5)(x - 5). Option A is only partially factored (didn't factor the difference of squares). Option C doesn't preserve the original expression. Option D only factors out 2, not the complete GCF.

Concept Tested: Factoring Completely

See: Factoring Completely

10. When factoring the trinomial 3x² + 11x + 6 using the AC method, what two numbers multiply to AC and add to b?

  1. 3 and 6
  2. 11 and 1
  3. 9 and 2
  4. 2 and 9
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The correct answer is D. In the AC method, AC = 3 × 6 = 18, and b = 11. We need two numbers that multiply to 18 and add to 11. The numbers 2 and 9 work: 2 × 9 = 18 and 2 + 9 = 11. Option A gives the original a and c values, not the numbers we need. Option B doesn't multiply to 18. Option C is the same as option D (order doesn't matter for this step).

Concept Tested: Factoring Trinomials

See: Factoring ax² + bx + c

Quiz Complete

Review the explanations above to reinforce your understanding of factoring polynomials. For additional practice, refer back to the chapter content.