Solve 4a^4-34a^2-18

$$4 a ^ { 4 } - 34 a ^ { 2 } - 18$$

Factor

$$2\left(a-3\right)\left(a+3\right)\left(2a^{2}+1\right)$$

Solution Steps

Factor out 2.

$$2\left(2a^{4}-17a^{2}-9\right)$$

Consider 2a^{4}-17a^{2}-9. Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 2a^{4} and n divides the constant factor -9. One such factor is 2a^{2}+1. Factor the polynomial by dividing it by this factor.

$$\left(2a^{2}+1\right)\left(a^{2}-9\right)$$

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

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

Rewrite the complete factored expression. Polynomial 2a^{2}+1 is not factored since it does not have any rational roots.

$$2\left(2a^{2}+1\right)\left(a-3\right)\left(a+3\right)$$

Evaluate

$$4a^{4}-34a^{2}-18$$