Question

Factor Sums and Differences of Cubes

In the following exercises, factor.

$a^3-125$

Short answer

The factors for$a^3-125=(a-5)(a^2+5a+25)$

Step-by-step solution

Step 1. Given information

The given differences of cubes $a^3-125$

Step 2. Finding the factors for given differences of cubes 

We can find the factors as follows

Use the sum product pattern

$$\begin{gathered} a^3-125=a^3-5^3[\because (a^3-b^3)=(a-b\left)\right(a^2+ab+b^2\left)\right] \\ =a^3+5a^2+25a-5a^2-25a-125 \end{gathered}$$

Common factors from the two pairs

$a^3+5a^2+25a-5a^2-25a-125=a(a^2+5a+25)-5(a^2+5a+25)$

Rewrite in factored form

$a(a^2+5a+25)-5(a^2+5a+25)=(a-5)(a^2+5a+25)$

Therefore,the factors for$a^3-125=(a-5)(a^2+5a+25)$