Solve m^4n^4+m^2n^2-132

$$m ^ { 4 } n ^ { 4 } + m ^ { 2 } n ^ { 2 } - 132$$

Factor

$$\left(m^{2}n^{2}-11\right)\left(m^{2}n^{2}+12\right)$$

Solution Steps

Consider m^{4}n^{4}+m^{2}n^{2}-132 as a polynomial over variable m.

$$n^{4}m^{4}+n^{2}m^{2}-132$$

Find one factor of the form n^{k}m^{p}+q, where n^{k}m^{p} divides the monomial with the highest power n^{4}m^{4} and q divides the constant factor -132. One such factor is m^{2}n^{2}+12. Factor the polynomial by dividing it by this factor.

$$\left(m^{2}n^{2}+12\right)\left(m^{2}n^{2}-11\right)$$

Evaluate

$$\left(mn\right)^{4}+\left(mn\right)^{2}-132$$