Solve x^3-2x^2-51x-108=0

$$x ^ { 3 } - 2 x ^ { 2 } - 51 x - 108 = 0$$

Solve for x

$$x=-3$$

$$x=9$$

$$x=-4$$

Solution Steps

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.

$$x=-3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-2x^{2}-51x-108 by x+3 to get x^{2}-5x-36. Solve the equation where the result equals to 0.

$$x^{2}-5x-36=0$$

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, -5 for b, and -36 for c in the quadratic formula.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\left(-36\right)}}{2}$$

Do the calculations.

$$x=\frac{5±13}{2}$$

Solve the equation x^{2}-5x-36=0 when ± is plus and when ± is minus.

$$x=-4 x=9$$

List all found solutions.

$$x=-3 x=-4 x=9$$