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Question

Difficulty: EasyFactoring Polynomials

What is the completely factored form of the polynomial 3x2−3x−363x^2 - 3x - 36?

  1. 3(x−4)(x+3)3(x - 4)(x + 3)Answer
  2. B
    3(x+4)(x−3)3(x + 4)(x - 3)
  3. C
    (3x−12)(x+3)(3x - 12)(x + 3)
  4. D
    3(x−6)(x+2)3(x - 6)(x + 2)

Answer

The completely factored form is 3(x−4)(x+3)3(x - 4)(x + 3).
The correct answer accurately extracts the greatest common factor of 33 first, and then properly factors the resulting quadratic x2−x−12x^2 - x - 12 into (x−4)(x+3)(x - 4)(x + 3).

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from all terms in the polynomial 3x2−3x−363x^2 - 3x - 36.
3(x2−x−12)3(x^2 - x - 12)
Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor.
2
Find two numbers that multiply to the constant term (−12-12) and add up to the coefficient of the middle term (−1-1).
The numbers are −4-4 and 33, because −4×3=−12-4 \times 3 = -12 and −4+3=−1-4 + 3 = -1.
These numbers are required to construct the binomial factors of the quadratic expression.
3
Write the final completely factored expression by combining the GCF and the two binomial factors.
3(x−4)(x+3)3(x - 4)(x + 3)
This represents the full decomposition of the original polynomial into its simplest multiplicative components.

Key Concept

Factoring out a greatest common factor (GCF) followed by factoring a basic quadratic trinomial.
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Topics

Factoring Polynomials

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