How do you factor completely #x^4 + 5x^2 - 36#?

1 Answer
May 1, 2016

Answer:

#x^4+5x^2-36#

#=(x^2+9)(x-2)(x+2)#

#=(x-3i)(x+3i)(x-2)(x+2)#

Explanation:

Use the difference of squares identity once or twice:

#a^2-b^2 = (a-b)(a+b)#

We can treat this as a quadratic in #x^2# first to find:

#x^4+5x^2-36#

#=(x^2+9)(x^2-4)#

#=(x^2+9)(x^2-2^2)#

#=(x^2+9)(x-2)(x+2)#

The remaining quadratic factor has no linear factors with Real coefficients, but it can also be treated as a difference of squares using #9 = -(3i)^2# as follows:

#=(x^2-(3i)^2)(x-2)(x+2)#

#=(x-3i)(x+3i)(x-2)(x+2)#