# Can't understand $a^3 + b^3 = (a+b)^3-3ab(a+b)$

I tried solving an exercise and was unsuccessful in doing so. I checked the solution and it had this piece $a^3 + b^3 = (a+b)^3-3ab(a+b)$.

Another exercise also had a similar formula: $a^2+b^2=(a+b)^2 - 2ab$

I can see a similar pattern but it's pointless to study the pattern if I don't understand it at all. Is this some kind of special product?

• Have you tried expanding the RHS? – Wyllich Aug 30 '17 at 14:29
• – Mauro ALLEGRANZA Aug 30 '17 at 14:30
• For the second case: $(a+b)^2= a^2+b^2+2ab$. – Mauro ALLEGRANZA Aug 30 '17 at 14:31
• You can try with some very simple examples: $(3+2)^2=5^2=25 \ne 13=3^2+2^2$ – Mauro ALLEGRANZA Aug 30 '17 at 14:51

## 3 Answers

expanding the right-hand side we get $$a^3+3a^2b+3ab^2+b^3-3a^2b-3ab^2=...$$

• Finally I see it, so if we expand $(a+b)^3$ we see an $a^3$ and a $b^3$ plus a whole lot extra. So you remove the extra. To end up again with $a^3 + b^3$ – user3051847 Aug 30 '17 at 14:39

$(a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+3ab(a+b)+b^3.$

A different way of thinking:

To take $3$ balls from $a$ and $b$, that are either all $3$ $a$ or all $3$ $b$, you can take any combination from $(a+b)$, but you have to subtract the options where both $a$ and $b$ occur. There are 3 such variations, one place out of three has a different symbol.