Solve x^4-33x^2-108

$${ x }^{ 4 } -33 { x }^{ 2 } -108$$

Factor

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

Solution Steps

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

$$x^{4}-33x^{2}-108=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -108 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±108,±54,±36,±27,±18,±12,±9,±6,±4,±3,±2,±1$$

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.

$$x=6$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-33x^{2}-108 by x-6 to get x^{3}+6x^{2}+3x+18. To factor the result, solve the equation where it equals to 0.

$$x^{3}+6x^{2}+3x+18=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 18 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±18,±9,±6,±3,±2,±1$$

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.

$$x=-6$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+6x^{2}+3x+18 by x+6 to get x^{2}+3. To factor the result, solve the equation where it equals to 0.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 0 for b, and 3 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$x=\frac{0±\sqrt{-12}}{2}$$

Polynomial x^{2}+3 is not factored since it does not have any rational roots.

$$x^{2}+3$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

$$x^{4}-33x^{2}-108$$