Question

what is the factored form of $250x^3 - 16$?

a. $2(5x - 2)(25x^2 + 10x + 4)$

b. $2(5x + 2)(25x^2 - 10x + 4)$

c. $(5x - 2)(25x^2 + 10x + 4)$

d. $(5x + 2)(25x^2 - 10x + 4)$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of \(250x^3\) and \(-16\). The GCF of 250 and 16 is 2. So, factor out 2:
\(250x^3 - 16 = 2(125x^3 - 8)\)

Step2: Recognize the difference of cubes

Notice that \(125x^3 = (5x)^3\) and \(8 = 2^3\). The expression \(125x^3 - 8\) is a difference of cubes, which has the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(a = 5x\) and \(b = 2\).

Step3: Apply the difference of cubes formula

Using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) with \(a = 5x\) and \(b = 2\):
\(125x^3 - 8 = (5x - 2)((5x)^2 + (5x)(2) + 2^2) = (5x - 2)(25x^2 + 10x + 4)\)

Step4: Combine with the GCF

Substitute back into the expression from Step1:
\(250x^3 - 16 = 2(5x - 2)(25x^2 + 10x + 4)\)

Answer:

A. \(2(5x - 2)(25x^2 + 10x + 4)\)