Solve 4a^4-15a^2b^2-81b^4

$$4 a ^ { 4 } - 15 a ^ { 2 } b ^ { 2 } - 81 b ^ { 4 }$$

Factor

$$\left(4a^{2}-27b^{2}\right)\left(a^{2}+3b^{2}\right)$$

Solution Steps

Consider 4a^{4}-15a^{2}b^{2}-81b^{4} as a polynomial over variable a.

$$4a^{4}-15b^{2}a^{2}-81b^{4}$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 4a^{4} and n divides the constant factor -81b^{4}. One such factor is 4a^{2}-27b^{2}. Factor the polynomial by dividing it by this factor.

$$\left(4a^{2}-27b^{2}\right)\left(a^{2}+3b^{2}\right)$$

Evaluate

$$4a^{4}-81b^{4}-15\left(ab\right)^{2}$$