Answer:
[tex]\frac{x^5+y^5}{x+y}=x^4-x^3y+x^2y^2-xy^3+y^4[/tex]
Step-by-step explanation:
Given the expression
[tex]\left(x^5+y^5\right)\div \left(x+y\right)[/tex]
[tex]\mathrm{Apply\:factoring\:rule:\:}x^n+y^n=\left(x+y\right)\left(x^{n-1}-x^{n-2}y+\:\dots \:-\:xy^{n-2}\:+\:y^{n-1}\right)\:\quad \quad \mathrm{n\:is\:odd}[/tex]
[tex]x^5+y^5=\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)[/tex]
[tex]=\frac{\left(x+y\right)\left(x^4-x^3y+x^2y^2-xy^3+y^4\right)}{x+y}[/tex]
Cancel the common factor: x+y
[tex]=x^4-x^3y+x^2y^2-xy^3+y^4[/tex]
Thus,
[tex]\frac{x^5+y^5}{x+y}=x^4-x^3y+x^2y^2-xy^3+y^4[/tex]
