Question

13 factor the following using polynomial identities 125x³ - 64 a (5x + 8)(5x - 8) b (5x - 8)² c (5x - 4)(25x² + 20x + 16) d (5x + 4)(25x² - 20x + 16)

Explanation:

Step1: Recall Difference of Cubes

The expression \(125x^3 - 64\) is a difference of cubes. The formula for \(a^3 - b^3\) is \((a - b)(a^2 + ab + b^2)\).

Step2: Identify \(a\) and \(b\)

Here, \(a^3 = 125x^3\), so \(a = 5x\) (since \((5x)^3 = 125x^3\)). And \(b^3 = 64\), so \(b = 4\) (since \(4^3 = 64\)).

Step3: Apply the Formula

Substitute \(a = 5x\) and \(b = 4\) into the difference of cubes formula:
\(a^3 - b^3=(a - b)(a^2 + ab + b^2)=(5x - 4)((5x)^2+(5x)(4)+4^2)\)
Simplify the second factor: \((5x)^2 = 25x^2\), \((5x)(4)=20x\), \(4^2 = 16\), so it becomes \(25x^2 + 20x + 16\).
Thus, \(125x^3 - 64=(5x - 4)(25x^2 + 20x + 16)\).

Answer:

C. \((5x - 4)(25x^2 + 20x + 16)\)