Solve h^3-125h+500=0

$$h ^ { 3 } - 125 h + 500 = 0$$

Solve for h

$$h=\frac{-5\sqrt{17}-5}{2}\approx -12.807764064$$

$$h=5$$

$$h = \frac{5 \sqrt{17} - 5}{2} \approx 7.807764064$$

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

$$±500,±250,±125,±100,±50,±25,±20,±10,±5,±4,±2,±1$$

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.

$$h=5$$

By Factor theorem, h-k is a factor of the polynomial for each root k. Divide h^{3}-125h+500 by h-5 to get h^{2}+5h-100. Solve the equation where the result equals to 0.

$$h^{2}+5h-100=0$$

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 -100 for c in the quadratic formula.

$$h=\frac{-5±\sqrt{5^{2}-4\times 1\left(-100\right)}}{2}$$

Do the calculations.

$$h=\frac{-5±5\sqrt{17}}{2}$$

Solve the equation h^{2}+5h-100=0 when ± is plus and when ± is minus.

$$h=\frac{-5\sqrt{17}-5}{2} h=\frac{5\sqrt{17}-5}{2}$$

List all found solutions.

$$h=5 h=\frac{-5\sqrt{17}-5}{2} h=\frac{5\sqrt{17}-5}{2}$$