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Difficulty: EasyFactoring Polynomials

Which of the following is the completely factored form of the expression x3−9xx^3 - 9x?

  1. A
    x(x−3)2x(x - 3)^2
  2. B
    x(x3−9)x(x^3 - 9)
  3. x(x−3)(x+3)x(x - 3)(x + 3)Answer
  4. D
    9x(x−1)(x+1)9x(x - 1)(x + 1)
  5. E
    x(x+3)2x(x + 3)^2

Answer

x(x−3)(x+3)x(x - 3)(x + 3)
The expression x3−9xx^3 - 9x can be factored by first finding the greatest common factor of the terms. Since both terms share a factor of xx, factoring out xx yields x(x2−9)x(x^2 - 9). The binomial x2−9x^2 - 9 is a difference of squares because it can be written as x2−32x^2 - 3^2. Applying the difference of squares formula, a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b), gives (x−3)(x+3)(x - 3)(x + 3). Therefore, the completely factored form is x(x−3)(x+3)x(x - 3)(x + 3).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) of the terms in the expression.
x(x2−9)x(x^2 - 9)
Both x3x^3 and 9x9x share a common factor of xx.
2
Factor the remaining binomial expression inside the parentheses using the difference of squares formula.
x(x−3)(x+3)x(x - 3)(x + 3)
The expression x2−9x^2 - 9 is a difference of squares, which factors as (a−b)(a+b)(a - b)(a + b) where a=xa = x and b=3b = 3.

Key Concept

Factoring out the greatest common factor and factoring a difference of squares.
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Topics

Factoring Polynomials

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