How do you factor #-x^3 + 15x^2 - 75x + 125#?

1 Answer
George C.
Jun 11, 2015

Spotted #-x^3+15x^2-75x+125 = (5-x)^3#

Explanation:

Noticing that the first term is #(-x)^3# and the last is #5^3#, I immediately considered the possibility that this is a perfect cube quadrinomial - in fact #(5-x)^3#.

In general #(a-b)^3 = a^3-3a^2b+3ab^2-b^3#

If we put #a = 5# and #b = x# then

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

#=125-75x+15x^2-x^3#

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