Solve 3x^6-3x^4+6

$$3 x ^ { 6 } - 3 x ^ { 4 } + 6$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(x^{6}-x^{4}+2\right)$$

Consider x^{6}-x^{4}+2. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{6} and m divides the constant factor 2. One such factor is x^{2}+1. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}+1,x^{4}-2x^{2}+2.

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

Evaluate

$$3\left(x^{6}-x^{4}+2\right)$$