Question

1. factor the following polynomial completely. if it cannot be factored, write prime polynomial.

27p³ + 8q³

Explanation:

Step1: Identify the formula for sum of cubes

The polynomial \(27p^{3}+8q^{3}\) is a sum of cubes. The formula for factoring the sum of cubes is \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\). Here, \(a^{3}=27p^{3}\), so \(a = \sqrt[3]{27p^{3}}=3p\), and \(b^{3}=8q^{3}\), so \(b=\sqrt[3]{8q^{3}} = 2q\).

Step2: Apply the sum of cubes formula

Substitute \(a = 3p\) and \(b = 2q\) into the formula \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\).
First, calculate \(a + b=3p + 2q\).
Then, calculate \(a^{2}=(3p)^{2}=9p^{2}\), \(ab=(3p)(2q)=6pq\), and \(b^{2}=(2q)^{2}=4q^{2}\).
So, \(a^{2}-ab + b^{2}=9p^{2}-6pq + 4q^{2}\).

Answer:

\((3p + 2q)(9p^{2}-6pq + 4q^{2})\)