- #1

brotherbobby

- 669

- 159

- Homework Statement
- If ##x+y=m## and ##x-y=n##, express ##\boldsymbol{x^3+y^3}## in terms of ##\boldsymbol m## and ##\boldsymbol n##.

- Relevant Equations
- We have ##a^3+b^3=(a+b)(a^2-ab+b^2)##

**Attempt :**

##{\color{red}{\large *}}## The key thing is to do the problem in a "good way", whereby I can express ##x^3+y^3=f\{(x+y), (x-y)\}## purely, with no cross terms - and that is not what I am able to do.

Below I reproduce what I could do, both methods less than satisfactory.

__Method 1 (bad - brute force)__: Given ##x+y=m## and ##x-y=n##, we solve them to yield ##x=\dfrac{m+n}{2}, y = \dfrac{m-n}{2}##.

\dfrac{m}{4}(m^2+3n^2)}\quad{\color{green}{\Large{\checkmark}}}##, agreeing with the answer in the text shown alongside.

__Method 2 (partly satisfactory)__: Let's write ##\small{x^3+y^3 = (x+y)(x^2-xy+y^2)=(x+y)\{(x-y)^2+xy\}=m\left\{n^2+\dfrac{1}{4}(m^2-n^2)\right\}=\dfrac{m}{4}(4n^2+m^2-n^2)=\boxed{\dfrac{m}{4}(m^2+3n^2}}\quad{\color{green}{\Large{\checkmark}}}##

**Request :**Is there a neater way to solve it, as I mentioned above alongside the ##{\color{red}{\large *}}## ?

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