How do you factor #(r+6)^3-216#?
1 Answer
May 2, 2016
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
#(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)#
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)#