Solve h^3-81h+154=0

$$h ^ { 3 } - 81 h + 154 = 0$$

Solve for h (complex solution)

$$h=\sqrt{78}-1\approx 7.831760866$$

$$h=-\left(\sqrt{78}+1\right)\approx -9.831760866$$

$$h=2$$

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

$$±154,±77,±22,±14,±11,±7,±2,±1$$

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.

$$h=2$$

By Factor theorem, h-k is a factor of the polynomial for each root k. Divide h^{3}-81h+154 by h-2 to get h^{2}+2h-77. Solve the equation where the result equals to 0.

$$h^{2}+2h-77=0$$

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, 2 for b, and -77 for c in the quadratic formula.

$$h=\frac{-2±\sqrt{2^{2}-4\times 1\left(-77\right)}}{2}$$

Do the calculations.

$$h=\frac{-2±2\sqrt{78}}{2}$$

Solve the equation h^{2}+2h-77=0 when ± is plus and when ± is minus.

$$h=-\sqrt{78}-1 h=\sqrt{78}-1$$

List all found solutions.

$$h=2 h=-\sqrt{78}-1 h=\sqrt{78}-1$$

Solve for h

$$h=\sqrt{78}-1\approx 7.831760866$$

$$h=-\sqrt{78}-1\approx -9.831760866$$

$$h=2$$

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

$$±154,±77,±22,±14,±11,±7,±2,±1$$

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.

$$h=2$$

By Factor theorem, h-k is a factor of the polynomial for each root k. Divide h^{3}-81h+154 by h-2 to get h^{2}+2h-77. Solve the equation where the result equals to 0.

$$h^{2}+2h-77=0$$

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, 2 for b, and -77 for c in the quadratic formula.

$$h=\frac{-2±\sqrt{2^{2}-4\times 1\left(-77\right)}}{2}$$

Do the calculations.

$$h=\frac{-2±2\sqrt{78}}{2}$$

Solve the equation h^{2}+2h-77=0 when ± is plus and when ± is minus.

$$h=-\sqrt{78}-1 h=\sqrt{78}-1$$

List all found solutions.

$$h=2 h=-\sqrt{78}-1 h=\sqrt{78}-1$$