Q. 6.62 - Intermediate Algebra

Question

Factor: $y^3+8$.

Step-by-Step Solution

Answer

Factors of $y^3+8$ are $\left(y+2\right)\left(y^2-2y+4\right)$.

Step 1. Given information.

Given an expression $y^3+8$.

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

It can be seen that $y^3+8=y^3+2^3$.

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.

$\begin{array}{rcl}{y}^3+8& =& y^3+2^3\\ & =& \left(y+2\right)\left(y^2-2y+4\right)\end{array}$

So, $y^3+8=\left(y+2\right)\left(y^2-2y+4\right)$.

Step 4. Check by multiplying the factors.

Verify as follows.

$\begin{array}{rcl}\left(y+2\right)\left(y^2-2y+4\right)& =& y\left(y^2-2y+4\right)+2\left(y^2-2y+4\right)\\ & =& y^3-2y^2+4y+2y^2-4y+8\\ & =& y^3+8\end{array}$