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Question

Difficulty: MediumFactoring Polynomials

Which of the following is the completely factored form of the expression 3x(x−2)2−12x3x(x - 2)^2 - 12x?

  1. A
    3x(x2−8)3x(x^2 - 8)
  2. B
    −9x(x−2)2-9x(x - 2)^2
  3. 3x2(x−4)3x^2(x - 4)Answer
  4. D
    3x(x−4)3x(x - 4)
  5. E
    12x(x−1)(x−3)12x(x - 1)(x - 3)

Answer

The completely factored form of the expression is 3x2(x−4)3x^2(x - 4).
The correct answer is 3x2(x−4)3x^2(x-4). Factoring out the greatest common factor 3x3x from the terms 3x(x−2)23x(x-2)^2 and −12x-12x gives 3x[(x−2)2−4]3x[(x-2)^2-4]. The expression within the brackets is a difference of squares that can be factored as [(x−2)−2][(x−2)+2][(x-2)-2][(x-2)+2], which simplifies to x(x−4)x(x-4). Multiplying this by the GCF 3x3x yields 3x2(x−4)3x^2(x-4).

Step-by-Step Solution

1
Factor out the greatest common factor (GCF), 3x3x, from both terms of the expression 3x(x−2)2−12x3x(x - 2)^2 - 12x.
3x[(x−2)2−4]3x[(x - 2)^2 - 4]
Both terms 3x(x−2)23x(x-2)^2 and 12x12x share the common factors 33 and xx.
2
Factor the difference of squares inside the bracket, (x−2)2−4(x-2)^2 - 4, using the formula a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b) where a=x−2a = x-2 and b=2b = 2.
[(x−2)−2][(x−2)+2]=(x−4)(x)[(x-2)-2][(x-2)+2] = (x-4)(x)
Since 4=224 = 2^2, the terms inside the brackets form a difference of squares.
3
Combine the factored parts and simplify the expression by multiplying the variable terms.
3x⋅x(x−4)=3x2(x−4)3x \cdot x(x-4) = 3x^2(x-4)
Multiplying 3x3x by xx requires adding their exponents (1+1=21 + 1 = 2).

Key Concept

Factoring polynomials using the greatest common factor (GCF) and difference of squares.
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Topics

Factoring Polynomials

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