Solve {l}{x^{4}-y^{4}+3x^2}{+3y^2+ax^2}{+ay^2}

$$\left. \begin{array} { l } { x ^ { 4 } - y ^ { 4 } + 3 x ^ { 2 } } \\ { + 3 y ^ { 2 } + a x ^ { 2 } } \\ { + a y ^ { 2 } } \end{array} \right.$$

Factor

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

Solution Steps

Consider x^{4}-y^{4}+3x^{2}+3y^{2}+ax^{2}+ay^{2} as a polynomial over variable x.

$$x^{4}+\left(3+a\right)x^{2}-y^{4}+3y^{2}+ay^{2}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor -y^{4}+ay^{2}+3y^{2}. One such factor is x^{2}+y^{2}. Factor the polynomial by dividing it by this factor.

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

Evaluate

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