Question

factor the polynomial.
27x³ - 64r³
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. 27x³ - 64r³ =

b. the polynomial cannot be factored.

Explanation:

Step1: Identify the formula for difference of cubes

The polynomial \(27x^{3}-64r^{3}\) is a difference of cubes. The formula for factoring a difference of cubes is \(a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})\).

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

For \(27x^{3}\), we have \(a^{3}=27x^{3}\), so \(a = \sqrt[3]{27x^{3}}=3x\). For \(64r^{3}\), we have \(b^{3}=64r^{3}\), so \(b=\sqrt[3]{64r^{3}} = 4r\).

Step3: Apply the difference of cubes formula

Substitute \(a = 3x\) and \(b = 4r\) into the formula \(a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})\):
\[

$$\begin{align*} 27x^{3}-64r^{3}&=(3x)^{3}-(4r)^{3}\\ &=(3x - 4r)((3x)^{2}+(3x)(4r)+(4r)^{2})\\ &=(3x - 4r)(9x^{2}+12xr + 16r^{2}) \end{align*}$$

\]

Answer:

A. \(27x^{3}-64r^{3}=(3x - 4r)(9x^{2}+12xr + 16r^{2})\)