How do you find all the zeros of F(x) = x^4 - 13x^2 + 36F(x)=x413x2+36?

1 Answer
Jul 16, 2016

Zeros: +-3±3, +-2±2

Explanation:

Since there are no terms of odd degree, we can treat this as a quadratic in x^2x2 to simplify it, then factorize each of the resulting quadratic factors.

First note that 36 = 9xx436=9×4 and 9+4=139+4=13, so we find:

x^4-13x^2+36x413x2+36

=(x^2)^2-13(x^2)+36=(x2)213(x2)+36

=(x^2-9)(x^2-4)=(x29)(x24)

=(x^2-3^2)(x^2-2^2)=(x232)(x222)

=(x-3)(x+3)(x-2)(x+2)=(x3)(x+3)(x2)(x+2)

Hence zeros:

+-3±3, +-2±2