Solve 36a^4+97a^2b^2+36b^4

$$36 { a }^{ 4 } +97 { a }^{ 2 } { b }^{ 2 } +36 { b }^{ 4 }$$

Factor

$$\left(4a^{2}+9b^{2}\right)\left(9a^{2}+4b^{2}\right)$$

Solution Steps

Consider 36a^{4}+97a^{2}b^{2}+36b^{4} as a polynomial over variable a.

$$36a^{4}+97b^{2}a^{2}+36b^{4}$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 36a^{4} and n divides the constant factor 36b^{4}. One such factor is 4a^{2}+9b^{2}. Factor the polynomial by dividing it by this factor.

$$\left(4a^{2}+9b^{2}\right)\left(9a^{2}+4b^{2}\right)$$

Evaluate

$$36a^{4}+36b^{4}+97\left(ab\right)^{2}$$