Solve x^4-7x^2-18

$$x ^ { 4 } - 7 x ^ { 2 } - 18$$

Factor

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

Solution Steps

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

$$x^{4}-7x^{2}-18=0$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-7x^{2}-18 by x-3 to get x^{3}+3x^{2}+2x+6. To factor the result, solve the equation where it equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±6,±3,±2,±1$$

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.

$$x=-3$$

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

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

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 2 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{0±\sqrt{-8}}{2}$$

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

$$x^{2}+2$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

$$x^{4}-7x^{2}-18$$