Solve 6x^6-3x^2-3

$$6 x ^ { 6 } - 3 x ^ { 2 } - 3$$

Factor

$$3\left(x-1\right)\left(x+1\right)\left(2x^{4}+2x^{2}+1\right)$$

Solution Steps

Factor out 3.

$$3\left(2x^{6}-x^{2}-1\right)$$

Consider 2x^{6}-x^{2}-1. Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 2x^{6} and n divides the constant factor -1. One such factor is x^{2}-1. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-1\right)\left(2x^{4}+2x^{2}+1\right)$$

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression. Polynomial 2x^{4}+2x^{2}+1 is not factored since it does not have any rational roots.

$$3\left(x-1\right)\left(x+1\right)\left(2x^{4}+2x^{2}+1\right)$$

Evaluate

$$6x^{6}-3x^{2}-3$$