How do you factor ${(2 x - 3)}^{3} - 27$ $(2x - 3)^3 - 27$ ?

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How do you factor ${(2 x - 3)}^{3} - 27$ $(2x - 3)^3 - 27$ ?

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Answer 1 · 2015-05-09T15:13:21

Factoring ${(2 x - 3)}^{3} - 27$ $(2x-3)^3 -27$

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

${(2 x - 3)}^{3} = 27$ $(2x-3)^3 = 27$

${(2 x - 3)}^{3} = 3^{3}$ $(2x-3)^3 = 3^3$

$2 x - 3 = 3$ $2x-3 = 3$

$x = 3$ $x=3$

So $(x - 3)$ $(x-3)$ is a factor of ${(2 x - 3)}^{3} - 27$ $(2x-3)^3-27$

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

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

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

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How do you factor ${(2 x - 3)}^{3} - 27$ $(2x - 3)^3 - 27$ ?

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How do you factor ${(2 x - 3)}^{3} - 27$ $(2x - 3)^3 - 27$ ?