Solve C(x)=x^4+x^2-y^4-y^2

$$C ( x ) = x ^ { 4 } + x ^ { 2 } - y ^ { 4 } - y ^ { 2 }$$

Factor

$$\left(x-y\right)\left(x+y\right)\left(x^{2}+y^{2}+1\right)$$

Solution Steps

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

$$x^{4}+x^{2}-y^{4}-y^{2}$$

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

$$\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}+1\right)$$

Consider x^{2}-y^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-y\right)\left(x+y\right)$$

Rewrite the complete factored expression.

$$\left(x^{2}+y^{2}+1\right)\left(x-y\right)\left(x+y\right)$$

Evaluate

$$\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}+1\right)$$