Question

1. factor the following polynomial completely. if it cannot be factored, write prime polynomial. 27p³ + 8q³

Explanation:

Step1: Identify the sum of cubes

The polynomial \(27p^3 + 8q^3\) can be written as \((3p)^3+(2q)^3\), since \(27p^3=(3p)^3\) and \(8q^3=(2q)^3\).

Step2: Apply the sum of cubes formula

The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). Here, \(a = 3p\) and \(b = 2q\).
Substituting these values into the formula, we get:
\((3p + 2q)((3p)^2-(3p)(2q)+(2q)^2)\)

Step3: Simplify the second factor

Simplify \((3p)^2-(3p)(2q)+(2q)^2\):
\((3p)^2 = 9p^2\), \((3p)(2q)=6pq\), and \((2q)^2 = 4q^2\).
So the second factor becomes \(9p^2-6pq + 4q^2\).

Answer:

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