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Question

Difficulty: MediumFactoring Polynomials

Which of the following represents the completely factored form of the quadratic expression 3x2−3x−183x^2 - 3x - 18?

  1. 3(x−3)(x+2)3(x - 3)(x + 2)Answer
  2. B
    3(x+3)(x−2)3(x + 3)(x - 2)
  3. C
    (3x−9)(x+2)(3x - 9)(x + 2)
  4. D
    3(x−6)(x+1)3(x - 6)(x + 1)

Answer

The completely factored form is 3(x−3)(x+2)3(x - 3)(x + 2).
The correct answer properly extracts the GCF of 3 first, leaving the trinomial x2−x−6x^2 - x - 6. This trinomial factors into (x−3)(x+2)(x - 3)(x + 2) because −3-3 and 22 multiply to −6-6 and add up to −1-1.

Step-by-Step Solution

1
Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression.
3(x2−x−6)3(x^2 - x - 6)
The coefficients 3,−3,3, -3, and −18-18 are all divisible by 33.
2
Find two numbers that multiply to the constant term (−6-6) and add to the coefficient of the middle term (−1-1).
The numbers are −3-3 and 22.
−3×2=−6-3 \times 2 = -6 and −3+2=−1-3 + 2 = -1.
3
Rewrite the quadratic trinomial as the product of two binomials using these numbers.
3(x−3)(x+2)3(x - 3)(x + 2)
This represents the final, completely factored form of the original expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a quadratic trinomial.
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Topics

Factoring Polynomials

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