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 \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\).
We get \((3p + 2q)((3p)^2-(3p)(2q)+(2q)^2)\).
Simplify the second factor: \((3p)^2 = 9p^2\), \((3p)(2q)=6pq\), \((2q)^2 = 4q^2\). So the second factor is \(9p^2-6pq + 4q^2\).

Answer:

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