Question

which of the following is the correct factorization of the polynomial below? 27x^3 + 64
a. (3x + 4)(9x^2 - 12x + 16)
b. (9x + 8)(3x^2 - 16x + 8)
c. (3x^2 + 8)(9x - 16x + 8)
d. the polynomial is irreducible.

Explanation:

Step1: Recall sum - of - cubes formula

The sum - of - cubes formula is $a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})$.

Step2: Identify a and b

For the polynomial $27x^{3}+64$, we have $a = 3x$ (since $(3x)^{3}=27x^{3}$) and $b = 4$ (since $4^{3}=64$).

Step3: Apply the formula

Substitute $a = 3x$ and $b = 4$ into the sum - of - cubes formula:
$(3x)^{3}+4^{3}=(3x + 4)((3x)^{2}-(3x)\times4+4^{2})=(3x + 4)(9x^{2}-12x + 16)$.

Answer:

A. $(3x + 4)(9x^{2}-12x + 16)$