How can you factor #f(x)=x^4-12x^3+59x^2-138x+130#?

1 Answer
George C.
Oct 17, 2017

#x^4-12x^3+59x^2-138x+130#

#=(x^2-6x+10)(x^2-6x+13)#

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

Explanation:

Given:

#f(x) = x^4-12x^3+59x^2-138x+130#

Simplify the quartic by noting the similarity to #(x-3)^4# ...

#x^4-12x^3+59x^2-138x+130#

#=x^4-12x^3+54x^2-108x+81+5x^2-30x+45+4#

#=(x-3)^4+5(x-3)^2+4#

#=((x-3)^2+1)((x-3)^2+4) color(grey)(= (x^2-6x+10)(x^2-6x+13))#

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

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

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