How do you factor completely #64 + a^3 #?

1 Answer
Bill K.
Jul 13, 2016

#a^3+64=(a+4)(a^2-4a+16)#

Explanation:

The expression #a^3+64=a^3+4^3# is the sum of two cubes (3rd powers). In general, if you have an arbitrary sum of two cubes, say #x^3+y^3#, it can be factored as:

#x^3+y^3=(x+y)(x^2-xy+y^2)#. You should check this by expansion (multiplication) of the right-hand side.

Now apply this formula to the problem at hand by substituting #x=a# and #y=4#.

The difference of two cubes can also be factored, and the formula for the sum of two cubes can be used to do it:

#x^3-y^3=x^3+(-y)^3#

#=(x+(-y))(x^2-x(-y)+(-y)^2)#

#=(x-y)(x^2+xy+y^2)#.

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