How do you factor (r+6)^3-216?

1 Answer
May 2, 2016

(r+6)^3-216=r(r^2+18r+108)

Explanation:

The difference of cubes identity can be written:

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

We can use this with a=(r+6) and b=6 as follows:

(r+6)^3-216

=(r+6)^3-6^3

=((r+6)-6)((r+6)^2+(r+6)(6)+6^2)

=r((r^2+12r+36)+(6r+36)+36)

=r(r^2+18r+108)

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Footnote

The remaining quadratic factor cannot be factorised further using Real coefficients.

It can be factorised by completing the square using Complex coefficients:

r^2+18r+108

=(r+9)^2-81+108

=(r+9)^2+27

=(r+9)^2-(3sqrt(3)i)^2

=((r+9)-3sqrt(3)i)((r+9)+3sqrt(3)i)

=(r+9-3sqrt(3)i)(r+9+3sqrt(3)i)