Solve x^3-3x^2-8x+4=0

$${ x }^{ 3 } -3 { x }^{ 2 } -8x+4=0$$

Solve for x

$$x = \frac{\sqrt{17} + 5}{2} \approx 4.561552813$$

$$x=\frac{5-\sqrt{17}}{2}\approx 0.438447187$$

$$x=-2$$

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

$$±4,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-3x^{2}-8x+4 by x+2 to get x^{2}-5x+2. Solve the equation where the result equals to 0.

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

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, -5 for b, and 2 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{5±\sqrt{17}}{2}$$

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

$$x=\frac{5-\sqrt{17}}{2} x=\frac{\sqrt{17}+5}{2}$$

List all found solutions.

$$x=-2 x=\frac{5-\sqrt{17}}{2} x=\frac{\sqrt{17}+5}{2}$$