Solve g(x)=-x^4+4x^2-3

$$g ( x ) = - x ^ { 4 } + 4 x ^ { 2 } - 3$$

Factor

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

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$-x^{4}+4x^{2}-3=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3 and q divides the leading coefficient -1. List all candidates \frac{p}{q}.

$$±3,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide -x^{4}+4x^{2}-3 by x+1 to get -x^{3}+x^{2}+3x-3. To factor the result, solve the equation where it equals to 0.

$$-x^{3}+x^{2}+3x-3=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3 and q divides the leading coefficient -1. List all candidates \frac{p}{q}.

$$±3,±1$$

To factor the expression, solve the equation where it equals to 0.

$$\text{true}$$

Rewrite the factored expression using the obtained roots. Polynomial x^{2}-3 is not factored since it does not have any rational roots.

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

Evaluate

$$-x^{4}+4x^{2}-3$$