Solve f(x)=-1/2x^4+1/2

$$f ( x ) = \frac { - 1 } { 2 } x ^ { 4 } + \frac { 1 } { 2 }$$

Factor

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

Solution Steps

Factor out \frac{1}{2}.

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

Consider -x^{4}+1. Rewrite -x^{4}+1 as 1^{2}-\left(-x^{2}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Reorder the terms.

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

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

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

Reorder the terms.

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

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

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

Evaluate

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

Solution Steps

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

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