How do you simplify x^2/(x^3-125)+5/(x^2+5x+25)?

Feb 4, 2015

The answer is: $\frac{{x}^{2} + 5 x - 25}{\left(x - 5\right) \left({x}^{2} + 5 x + 25\right)}$.

First of all, we have to factor all the denominators.

${x}^{3} - 125 = {x}^{3} - {5}^{3} = \left(x - 5\right) \left({x}^{2} + 5 x + 25\right)$, with the polynomial ${x}^{2} + 5 x + 25$ no more factored.

I remember that ${a}^{3} - {b}^{3}$ are called "difference of cubes", and it could be factored:

${a}^{3} - {b}^{3} = \left(a + b\right) \left({a}^{2} - a b + {b}^{2}\right)$.

So:

${x}^{2} / \left(\left(x - 5\right) \left({x}^{2} + 5 x + 25\right)\right) + \frac{5}{{x}^{2} + 5 x + 25} =$

$= \frac{{x}^{2} + 5 \left(x - 5\right)}{\left(x - 5\right) \left({x}^{2} + 5 x + 25\right)} = \frac{{x}^{2} + 5 x - 25}{\left(x - 5\right) \left({x}^{2} + 5 x + 25\right)}$