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Difficulty: MediumFactoring Polynomials

To analyze the zeroes of a mathematical function, the algebraic expression 3x3+6x2−9x3x^3 + 6x^2 - 9x must be written as a product of its simplest linear factors. Which of the following represents the completely factored equivalent of this expression?

  1. A
    3x(x−3)(x+1)3x(x - 3)(x + 1)
  2. B
    x(3x+9)(x−1)x(3x + 9)(x - 1)
  3. 3x(x+3)(x−1)3x(x + 3)(x - 1)Answer
  4. D
    3x2(x−1)3x^2(x - 1)

Answer

The completely factored equivalent is 3x(x+3)(x−1)3x(x + 3)(x - 1).
The correct answer successfully extracts the greatest common factor, 3x3x, leaving the quadratic trinomial x2+2x−3x^2 + 2x - 3. This trinomial factors correctly into (x+3)(x−1)(x + 3)(x - 1) because 33 and −1-1 multiply to −3-3 and add to positive 22.

Step-by-Step Solution

1
Identify and extract the Greatest Common Factor (GCF) from all three terms in the expression 3x3+6x2−9x3x^3 + 6x^2 - 9x.
3x(x2+2x−3)3x(x^2 + 2x - 3)
Factoring out the GCF simplifies the remaining polynomial, making it possible to factor the quadratic trinomial.
2
Find two integers that multiply to the constant term (−3-3) and add to the middle coefficient (22).
The correct integers are 33 and −1-1.
These values are needed to factor the quadratic trinomial x2+2x−3x^2 + 2x - 3 into two binomials.
3
Rewrite the trinomial as a product of two binomials and combine it with the previously extracted GCF.
3x(x+3)(x−1)3x(x + 3)(x - 1)
This provides the final, completely factored form of the original algebraic expression.

Key Concept

Factoring Polynomials via GCF and Trinomial Methods
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Topics

Factoring Polynomials

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