Solve a^4b^4+6a^2b^2-55

$$a ^ { 4 } b ^ { 4 } + 6 a ^ { 2 } b ^ { 2 } - 55$$

Factor

$$\left(a^{2}b^{2}-5\right)\left(a^{2}b^{2}+11\right)$$

Solution Steps

Consider a^{4}b^{4}+6a^{2}b^{2}-55 as a polynomial over variable a.

$$b^{4}a^{4}+6b^{2}a^{2}-55$$

Find one factor of the form b^{k}a^{m}+n, where b^{k}a^{m} divides the monomial with the highest power b^{4}a^{4} and n divides the constant factor -55. One such factor is a^{2}b^{2}+11. Factor the polynomial by dividing it by this factor.

$$\left(a^{2}b^{2}+11\right)\left(a^{2}b^{2}-5\right)$$

Evaluate

$$\left(\left(ab\right)^{2}-5\right)\left(\left(ab\right)^{2}+11\right)$$