Q 6.61 - Intermediate Algebra

Question

Factor: $x^3+27$.

Step-by-Step Solution

Answer

Factors of $x^3+27$ are $\left(x+3\right)\left(x^2-3x+9\right)$.

Step 1. Given information.

Given an expression $x^3+27$.

Step 2. Check if it is a sum or difference of cubes pattern.

It can be seen that $x^3+27=x^3+3^3$.

The expression is sum of cubes pattern.

Step 3. Use either the sum or difference of cubes pattern and simplify.

Note that $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$.

The expression is sum of cubes.

$\begin{array}{rcl}{x}^3+27& =& x^3+3^3\\ & =& \left(x+3\right)\left(x^2-3x+9\right)\end{array}$

So, $x^3+27=\left(x+3\right)\left(x^2-3x+9\right)$.

Step 4. Check by multiplying the factors.

Verify as follows.$\begin{array}{rcl}\left(x+3\right)\left(x^2-3x+9\right)& =& x\left(x^2-3x+9\right)+3\left(x^2-3x+9\right)\\ & =& x^3-3x^2+9x+3x^2-9x+27\\ & =& x^3+27\end{array}$