Solve 9a^4-34a^2b^2+25b^4

$$9 { a }^{ 4 } -34 { a }^{ 2 } { b }^{ 2 } +25 { b }^{ 4 }$$

Factor

$$\left(a-b\right)\left(3a-5b\right)\left(a+b\right)\left(3a+5b\right)$$

Solution Steps

Consider 9a^{4}-34a^{2}b^{2}+25b^{4} as a polynomial over variable a.

$$9a^{4}-34b^{2}a^{2}+25b^{4}$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 9a^{4} and n divides the constant factor 25b^{4}. One such factor is 9a^{2}-25b^{2}. Factor the polynomial by dividing it by this factor.

$$\left(9a^{2}-25b^{2}\right)\left(a^{2}-b^{2}\right)$$

Consider 9a^{2}-25b^{2}. Rewrite 9a^{2}-25b^{2} as \left(3a\right)^{2}-\left(5b\right)^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(3a-5b\right)\left(3a+5b\right)$$

Consider a^{2}-b^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a-b\right)\left(a+b\right)$$

Rewrite the complete factored expression.

$$\left(a-b\right)\left(a+b\right)\left(3a-5b\right)\left(3a+5b\right)$$

Evaluate

$$9a^{4}+25b^{4}-34\left(ab\right)^{2}$$