Solve 1/2x^4-4x^2+6

$$\frac { 1 } { 2 } x ^ { 4 } - 4 x ^ { 2 } + 6$$

Factor

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

Solution Steps

Factor out \frac{1}{2}.

$$\frac{x^{4}-8x^{2}+12}{2}$$

Consider x^{4}-8x^{2}+12. 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 12. One such factor is x^{2}-6. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}-6,x^{2}-2.

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

Evaluate

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