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Question

Difficulty: MediumFactoring Polynomials

What is the completely factored form of the polynomial 3x2−12x−363x^2 - 12x - 36?

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

Answer

3(x−6)(x+2)3(x - 6)(x + 2)
To completely factor the polynomial, the greatest common factor must be extracted first. Factoring out the GCF of 3 yields 3(x2−4x−12)3(x^2 - 4x - 12). Next, factoring the trinomial x2−4x−12x^2 - 4x - 12 requires two integers that multiply to −12-12 and sum to −4-4. These integers are −6-6 and 22. Therefore, the completely factored form is 3(x−6)(x+2)3(x - 6)(x + 2).

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from all terms in the polynomial.
3(x2−4x−12)3(x^2 - 4x - 12)
Extracting the GCF simplifies the expression, making the quadratic trinomial easier to factor.
2
Find two numbers that multiply to the constant term (−12-12) and add to the coefficient of the middle term (−4-4).
The numbers are −6-6 and 22, because (−6)×2=−12(-6) \times 2 = -12 and (−6)+2=−4(-6) + 2 = -4.
These values are required to correctly split the middle term and factor the quadratic trinomial.
3
Write the completely factored form by combining the GCF and the two binomial factors.
3(x−6)(x+2)3(x - 6)(x + 2)
This represents the polynomial broken down into its simplest, irreducible algebraic factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.
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Topics

Factoring Polynomials

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