Solve left(x^3+y^3right)^+left(x^2+2xy+{y}^2right)

$${ \left( { x }^{ 3 } + { y }^{ 3 } \right) }^{ } + \left( { x }^{ 2 } +2xy+ { y }^{ 2 } \right)$$

Factor

$$\left(x+y\right)\left(x^{2}-xy+x+y^{2}+y\right)$$

Solution Steps

Multiply and combine like terms.

$$x^{3}+x^{2}+2xy+y^{3}+y^{2}$$

Consider x^{3}+x^{2}+2xy+y^{3}+y^{2} as a polynomial over variable x.

$$x^{3}+x^{2}+2yx+y^{3}+y^{2}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{3} and m divides the constant factor y^{3}+y^{2}. One such factor is x+y. Factor the polynomial by dividing it by this factor.

$$\left(x+y\right)\left(x^{2}-xy+x+y^{2}+y\right)$$

Evaluate

$$\left(x+y\right)\left(x^{2}-xy+x+y^{2}+y\right)$$

Solution Steps

Calculate x^{3}+y^{3} to the power of 1 and get x^{3}+y^{3}.

$$x^{3}+y^{3}+x^{2}+2xy+y^{2}$$