Question

question 5 of 10
which of the following is the correct factorization of the polynomial below?
27x³ + 64

a. (3x + 4)(9x² - 12x + 16)

b. (9x + 8)(3x² - 16x + 8)

c. (3x² + 8)(9x - 16x + 8)

d. the polynomial is irreducible.

Explanation:

Step1: Recall Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\).
Identify \(a\) and \(b\) in \(27x^3+64\). We have \(27x^3=(3x)^3\) and \(64 = 4^3\), so \(a = 3x\), \(b = 4\).

Step2: Apply the Formula

Substitute \(a = 3x\) and \(b = 4\) into the sum of cubes formula:
\((3x)^3+4^3=(3x + 4)((3x)^2-(3x)(4)+4^2)\)
Simplify each part: \((3x)^2 = 9x^2\), \((3x)(4)=12x\), \(4^2 = 16\).
So we get \((3x + 4)(9x^2-12x + 16)\), which matches option A.

Answer:

A. \((3x + 4)(9x^2 - 12x + 16)\)