Question 131915

{{{(s+p)(s^2-sp+p^2)}}} Start with the given expression




{{{s(s^2-sp+p^2)+p(s^2-sp+p^2)}}} Expand the expression. Remember something like {{{(a+b)(c+d+e)}}} expands to {{{a(c+d+e)+b(c+d+e)}}}



{{{(s)*(s^2)+(s)*(-sp)+(s)*(p^2)+(p)*(s^2)+(p)*(-sp)+(p)*(p^2)}}} Distribute



{{{s^3-s^2p+sp^2+ps^2-p^2s+p^3}}} Multiply



{{{s^3+ps^2-ps^2+p^2s-p^2s+p^3}}} Rearrange the terms



{{{s^3+cross(ps^2-ps^2)+cross(p^2s-p^2s)+p^3}}} Combine like terms. Notice how the inner terms cancel out



{{{s^3+p^3}}} Simplify 




So {{{(s+p)(s^2-sp+p^2)}}} expands and simplifies to {{{s^3+p^3}}}.


In other words, {{{(s+p)(s^2-sp+p^2)=s^3+p^3}}}