SOLUTION: Is (a+b)^3 ever equal to a^3+b^3? Explain.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Is (a+b)^3 ever equal to a^3+b^3? Explain.      Log On


   

Question 103883This question is from textbook Beginning Algebra
: Is (a+b)^3 ever equal to a^3+b^3? Explain. This question is from textbook Beginning Algebra

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let's work it out
%28a%2Bb%29%5E3=%28a%2Bb%29%2A%28a%2Bb%29%5E2
%28a%2Bb%29%5E3=%28a%2Bb%29%2A%28a%5E2%2B2ab%2Bb%5E2%29
%28a%2Bb%29%5E3=a%2A%28a%5E2%2B2ab%2Bb%5E2%29%2Bb%2A%28a%5E2%2B2ab%2Bb%5E2%29
%28a%2Bb%29%5E3=%28a%5E3%2B2a%5E2b%2Bab%5E2%29%2B%28ba%5E2%2B2ab%5E2%2Bb%5E3%29
%28a%2Bb%29%5E3=a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3
Now let's substitute.
%28a%2Bb%29%5E3=a%5E3%2Bb%5E3 Your original equation.
cross%28a%5E3%29%2B3a%5E2b%2B3ab%5E2%2Bcross%28b%5E3%29=cross%28a%5E3%29%2Bcross%28b%5E3%29 Substitute for the expanded cubic amd remove like terms from both sides.
3a%5E2b%2B3ab%5E2=0 Simplify.
%283a%5E2b%2B3ab%5E2%29%2F%283ab%29=0 Divide both sides by 3ab
%283a%5E2b%29%2F3ab%2B%283ab%5E2%29%2F3ab=0 Distribute.
a%2Bb=0 Simplify
So whenever a+b=0, then you know that
%28a%2Bb%29%5E3=a%5E3%2Bb%5E3
For example,
a=3 and b=-3
a%2Bb=0 and %28a%2Bb%29%5E3=0
a%5E3%2Bb%5E3=%2827%29%2B%28-27%29=0