# How do you factor v^4+71v^2+70?

May 17, 2015

Notice that $71 = 70 + 1$ and $70 = 70 \times 1$.

So we can factor ${v}^{4} + 71 {v}^{2} + 70 = \left({v}^{2} + 70\right) \left({v}^{2} + 1\right)$

This is an instance of the identity:
$\left(x + a\right) \left(x + b\right) = {x}^{2} + \left(a + b\right) x + \left(a \times b\right)$

With $x = {v}^{2}$, $a = 70$ and $b = 1$.

There are no linear factors with real coefficients because

${v}^{2} + 70 \ge 70 > 0$ and ${v}^{2} + 1 \ge 1 > 0$ for all real values of $v$.