How do you factor #(2x - 3)^3 - 27#?

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Answer 1 · 2015-05-09T15:13:21

Factoring #(2x-3)^3 -27#

Consider a related problem:
Find the solutions for #(2x-3)^3 -27 = 0#

#(2x-3)^3 = 27#

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

#2x-3 = 3#

#x=3#

So #(x-3)# is a factor of #(2x-3)^3-27#

Using synthetic division we can determine that
#(2x-3)^3-27 = (x-3)(8x^2-12x+18)#

We can extract a common factor of #(2)# from this final factor
#(2x-3)^3-27 = (x-3)(2)(4x^2-6x+9)#

Examining the discriminant of #(4x^2-6x+9)# reveals that there are no further factors.