Question

How do you divide `(x^5 + y^5) /(x + y)`?

Answer 1

For `n` odd integer the following identity holds

`x^n + y^n = (x + y)(x^{n-1} - x^{n-2}y + \cdots - xy^{n-2} + y^{n-1} )`

or

`[x^n + y^n]/[x+y]=(x^{n-1} - x^{n-2}y + \cdots - xy^{n-2} + y^{n-1} )`

Hence for our case we have that

`[x^5+y^5]/[x+y]=x^4-x^3 y+x^2 y^2-x y^3+y^4`