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

1 Answer
Jun 24, 2017

Answer:

#2x^3-125 = (root(3)(2)x-5)(root(3)(4)x^2+5root(3)(2)x+25)#

Explanation:

The difference of cubes identity can be written:

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

Note that #125 = 5^3# is a perfect cube (over the rationals), but #2x^3# is not.

We can treat it as a cube by using irrational coefficients, to find:

#2x^3 = (root(3)(2)x)^3#

and hence:

#2x^3-125 = (root(3)(2)x)^3-5^3#

#color(white)(2x^3-125) = (root(3)(2)x-5)((root(3)(2)x)^2+(root(3)(2)x)(5)+5^2)#

#color(white)(2x^3-125) = (root(3)(2)x-5)(root(3)(4)x^2+5root(3)(2)x+25)#

...noting that we have used #root(3)(a)root(3)(b) = root(3)(ab)# and hence:

#(root(3)(2))^2 = root(3)(2^2) = root(3)(4)#