# How do you know if 5y^4+10y^2+5 is a perfect square trinomial and how do you factor it?

Jun 7, 2015

It is a perfect square trinomial, it is of the form

${a}^{2} + 2 a b + {b}^{2} = {\left(a + b\right)}^{2}$, with $a = \sqrt{5} {y}^{2}$ and $b = \sqrt{5}$.

So we can write:

$5 {y}^{4} + 10 {y}^{2} + 5 = {\left(\sqrt{5} {y}^{2} + \sqrt{5}\right)}^{2}$

Generally it's nicer to use rational coefficients if possible and write instead:

$5 {y}^{4} + 10 {y}^{2} + 5 = 5 {\left({y}^{2} + 1\right)}^{2}$

$\left({y}^{2} + 1\right)$ has no simpler factors with real coefficients because:

${y}^{2} + 1 \ge 1 > 0$ for all $y \in \mathbb{R}$