Solve m^3-14m+8=0

$${ m }^{ 3 } -14m+8 = 0$$

Solve for m

$$m=\sqrt{2}+2\approx 3.414213562$$

$$m=2-\sqrt{2}\approx 0.585786438$$

$$m=-4$$

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

$$±8,±4,±2,±1$$

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.

$$m=-4$$

By Factor theorem, m-k is a factor of the polynomial for each root k. Divide m^{3}-14m+8 by m+4 to get m^{2}-4m+2. Solve the equation where the result equals to 0.

$$m^{2}-4m+2=0$$

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, -4 for b, and 2 for c in the quadratic formula.

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

Do the calculations.

$$m=\frac{4±2\sqrt{2}}{2}$$

Solve the equation m^{2}-4m+2=0 when ± is plus and when ± is minus.

$$m=2-\sqrt{2} m=\sqrt{2}+2$$

List all found solutions.

$$m=-4 m=2-\sqrt{2} m=\sqrt{2}+2$$