# How do you factor completely x^2(a^2+2ab+b^2) = x(a+b)?

Jun 16, 2015

You can first factor $\left({a}^{2} + 2 a b + {b}^{2}\right) = {\left(a + b\right)}^{2}$

#### Explanation:

The left part can then be rewritten:
${x}^{2} \cdot {\left(a + b\right)}^{2} = {\left[x \left(a + b\right)\right]}^{2}$
The complete equation changes into:
${\left[x \left(a + b\right)\right]}^{2} = x \left(a + b\right)$

This is of the form ${Q}^{2} = Q$ and there are only two solutions:
$Q = 0 \mathmr{and} Q = 1$

For $Q = 0 \to x = 0 \mathmr{and} \left(a + b\right) = 0 \to a = - b$

For $Q = 1 \to x \cdot \left(a + b\right) = 1 \to x = \frac{1}{a + b}$