can you factor 27a^3+64

Question

Can you factor 27a squared plus 64? Or is it already factored out?

Answer 1

27a³+64 is of the type x³+y³=(x+y)(x²-xy+y²).

So x³=27a³, making x=3a; and y³=64, making y=4.

27a³+64=(3a+4)(9a²-12a+16).

It factorises further but the factors are complex (not real):

x²-xy+y²=(x-½y+½iy√3)(x-½y-½iy√3) where i=√-1.

9a²-12a+16=(3a-2+2i√3)(3a-2-2i√3)=9a²-12a+4+12=9a²-12a+16.