Solve x^4+9x^2y^2+14y^4

$$x ^ { 4 } + 9 x ^ { 2 } y ^ { 2 } + 14 y ^ { 4 }$$

Factor

$$\left(x^{2}+2y^{2}\right)\left(x^{2}+7y^{2}\right)$$

Solution Steps

Consider x^{4}+9x^{2}y^{2}+14y^{4} as a polynomial over variable x.

$$x^{4}+9y^{2}x^{2}+14y^{4}$$

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 14y^{4}. One such factor is x^{2}+7y^{2}. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}+7y^{2}\right)\left(x^{2}+2y^{2}\right)$$

Evaluate

$$x^{4}+14y^{4}+9\left(xy\right)^{2}$$