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Question

Difficulty: HardFactoring Polynomials

Which of the following is a factor of the polynomial 8x3−12x2−18x+278x^3 - 12x^2 - 18x + 27?

  1. A
    2x−92x - 9
  2. B
    4x−94x - 9
  3. C
    4x−34x - 3
  4. 2x+32x + 3Answer
  5. E
    2x+92x + 9

Answer

The correct answer is 2x+32x + 3.
The correct answer is the factor 2x+32x + 3. By grouping the first two terms and the last two terms, we get 4x2(2x−3)−9(2x−3)4x^2(2x - 3) - 9(2x - 3). Factoring out the common binomial yields (4x2−9)(2x−3)(4x^2 - 9)(2x - 3). The term 4x2−94x^2 - 9 is a difference of squares, which factors into (2x−3)(2x+3)(2x - 3)(2x + 3). Therefore, the fully factored expression is (2x−3)2(2x+3)(2x - 3)^2(2x + 3), making the linear expression with a positive constant term the correct factor.

Step-by-Step Solution

1
Group the four terms of the polynomial into two pairs.
(8x3−12x2)−(18x−27)(8x^3 - 12x^2) - (18x - 27)
Grouping terms allows us to find common binomial factors in a cubic polynomial.
2
Factor out the greatest common factor (GCF) from each group.
4x2(2x−3)−9(2x−3)4x^2(2x - 3) - 9(2x - 3)
The GCF of 8x38x^3 and 12x212x^2 is 4x24x^2, and the GCF of −18x-18x and 2727 is −9-9 (factoring out the negative to match the binomials).
3
Factor out the common binomial factor (2x−3)(2x - 3).
(4x2−9)(2x−3)(4x^2 - 9)(2x - 3)
Both groups share the factor (2x−3)(2x - 3), so it can be factored out.
4
Factor the quadratic difference of squares (4x2−9)(4x^2 - 9).
(2x−3)(2x+3)(2x−3)=(2x−3)2(2x+3)(2x - 3)(2x + 3)(2x - 3) = (2x - 3)^2(2x + 3)
The term 4x2−94x^2 - 9 is a difference of squares of the form a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b), where a=2xa = 2x and b=3b = 3.

Key Concept

Factoring a cubic polynomial completely by grouping and then applying the difference of squares formula.
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Topics

Factoring Polynomials

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