Solve x^4-8x^2-33-14y^2-y^4

$$x ^ { 4 } - 8 x ^ { 2 } - 33 - 14 y ^ { 2 } - y ^ { 4 }$$

Factor

$$\left(x^{2}-y^{2}-11\right)\left(x^{2}+y^{2}+3\right)$$

Solution Steps

Consider x^{4}-8x^{2}-33-14y^{2}-y^{4} as a polynomial over variable x.

$$x^{4}-8x^{2}-33-14y^{2}-y^{4}$$

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}-14y^{2}-33. One such factor is x^{2}+y^{2}+3. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}+y^{2}+3\right)\left(x^{2}-y^{2}-11\right)$$

Evaluate

$$\left(x^{2}-y^{2}-11\right)\left(x^{2}+y^{2}+3\right)$$