How do you factor $v^{4} + 71 v^{2} + 70$ $v^4+71v^2+70$ ?

Question

1662 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-05-17T18:27:13

Notice that $71 = 70 + 1$ $71 = 70 + 1$ and $70 = 70 \times 1$ $70 = 70xx1$ .

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

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

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

There are no linear factors with real coefficients because

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

How do you factor v^4+71v^2+70v4+71v2+70v^4+71v^2+70?