Solve 5x^4-10x^2-315

$$5 x ^ { 4 } - 10 x ^ { 2 } - 315$$

Factor

$$5\left(x-3\right)\left(x+3\right)\left(x^{2}+7\right)$$

Solution Steps

Factor out 5.

$$5\left(x^{4}-2x^{2}-63\right)$$

Consider x^{4}-2x^{2}-63. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor -63. One such factor is x^{2}-9. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-9\right)\left(x^{2}+7\right)$$

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

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

Rewrite the complete factored expression. Polynomial x^{2}+7 is not factored since it does not have any rational roots.

$$5\left(x-3\right)\left(x+3\right)\left(x^{2}+7\right)$$

Evaluate

$$5\left(x^{4}-2x^{2}-63\right)$$