Simplifying this rational expression into simplest form?

Question

The expression: $\frac{5(x^2 - y^2)}{x^2 - 2xy + y^2} $

I factored the polynomial at the top, and got this: $\frac{5(x - y)(x+y)}{x^2 - 2xy + y^2} $

This is how far I've gone... how should I factor the bottom? I need some brushing up since I'm not very good at factoring!

Answer 1

You can factor the bottom:

$$x^2-2xy+y^2=(x-y)^2$$

Remember that:

$$(a\pm b)^2=a^2\pm2ab+b^2$$

Answer 2

$$\frac{5(x^2 - y^2)}{x^2 - 2xy + y^2} = \frac{5(x^2 - y^2)}{(x-y)^2}=$$

$$\frac{5(x- y)(x+y)}{(x-y)(x-y)}=\frac{5(x+y)}{(x-y)}$$

Answer 3

Solve the quadratic equation for $x$: $$\begin{align} x^2-2yx+y^2=0 \\ x_{1,2}=&\frac{\left(-\frac{b}{2}\right)\pm \sqrt{\left(-\frac{b}{2}\right)^2-ac}}{a}= \\ &\frac{\left(-\frac{-2y}{2}\right)\pm \sqrt{\left(-\frac{-2y}{2}\right)^2-1\cdot y^2}}{1}= \\ &y \Rightarrow \end{align} \\ x^2-2xy+y^2=(x-y)(x-y).$$ Alternatively, you can guess the roots of the quadratics: $$x^2-2yx+y^2=(x-a)(x-b) \Rightarrow \begin{cases} ab=y^2 \\ a+b=2y\end{cases} \Rightarrow \begin{cases} a=y \\ b=y \end{cases}.$$