Solve x^3+10x^2+31x-110=0

$$x ^ { 3 } + 10 x ^ { 2 } + 31 x - 110 = 0$$

Solve for x (complex solution)

$$x=-\sqrt{19}i-6\approx -6-4.358898944i$$

$$x=2$$

$$x=-6+\sqrt{19}i\approx -6+4.358898944i$$

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 -110 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±110,±55,±22,±11,±10,±5,±2,±1$$

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}+10x^{2}+31x-110 by x-2 to get x^{2}+12x+55. Solve the equation where the result equals to 0.

$$x^{2}+12x+55=0$$

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, 12 for b, and 55 for c in the quadratic formula.

$$x=\frac{-12±\sqrt{12^{2}-4\times 1\times 55}}{2}$$

Do the calculations.

$$x=\frac{-12±\sqrt{-76}}{2}$$

Solve the equation x^{2}+12x+55=0 when ± is plus and when ± is minus.

$$x=-\sqrt{19}i-6 x=-6+\sqrt{19}i$$

List all found solutions.

$$x=2 x=-\sqrt{19}i-6 x=-6+\sqrt{19}i$$

Solve for x

$$x=2$$

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 -110 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±110,±55,±22,±11,±10,±5,±2,±1$$

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}+10x^{2}+31x-110 by x-2 to get x^{2}+12x+55. Solve the equation where the result equals to 0.

$$x^{2}+12x+55=0$$

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, 12 for b, and 55 for c in the quadratic formula.

$$x=\frac{-12±\sqrt{12^{2}-4\times 1\times 55}}{2}$$

Do the calculations.

$$x=\frac{-12±\sqrt{-76}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=2$$