Solve x^3-3x^2-54x+216=0

$$x ^ { 3 } - 3 x ^ { 2 } - 54 x + 216 = 0$$

Solve for x

$$x = \frac{3 \sqrt{17} - 3}{2} \approx 4.684658438$$

$$x=\frac{-3\sqrt{17}-3}{2}\approx -7.684658438$$

$$x=6$$

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 216 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±216,±108,±72,±54,±36,±27,±24,±18,±12,±9,±8,±6,±4,±3,±2,±1$$

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^{3}-3x^{2}-54x+216 by x-6 to get x^{2}+3x-36. Solve the equation where the result equals to 0.

$$x^{2}+3x-36=0$$

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

$$x=\frac{-3±\sqrt{3^{2}-4\times 1\left(-36\right)}}{2}$$

Do the calculations.

$$x=\frac{-3±3\sqrt{17}}{2}$$

Solve the equation x^{2}+3x-36=0 when ± is plus and when ± is minus.

$$x=\frac{-3\sqrt{17}-3}{2} x=\frac{3\sqrt{17}-3}{2}$$

List all found solutions.

$$x=6 x=\frac{-3\sqrt{17}-3}{2} x=\frac{3\sqrt{17}-3}{2}$$