Solve {l}{x^3-y^3+3x^2}{+3xy+3y^2}

$$\left. \begin{array} { l } { x ^ { 3 } - y ^ { 3 } + 3 x ^ { 2 } } \\ { + 3 x y + 3 y ^ { 2 } } \end{array} \right.$$

Factor

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

Solution Steps

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

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

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}+3y^{2}. One such factor is x-y+3. Factor the polynomial by dividing it by this factor.

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

Evaluate

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