Solve x^3+x^2+11x+51=0

$$x ^ { 3 } + x ^ { 2 } + 11 x + 51 = 0$$

Solve for x (complex solution)

$$x=1-4i$$

$$x=-3$$

$$x=1+4i$$

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

$$±51,±17,±3,±1$$

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.

$$x=-3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+x^{2}+11x+51 by x+3 to get x^{2}-2x+17. Solve the equation where the result equals to 0.

$$x^{2}-2x+17=0$$

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 17 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\times 17}}{2}$$

Do the calculations.

$$x=\frac{2±\sqrt{-64}}{2}$$

Solve the equation x^{2}-2x+17=0 when ± is plus and when ± is minus.

$$x=1-4i x=1+4i$$

List all found solutions.

$$x=-3 x=1-4i x=1+4i$$

Solve for x

$$x=-3$$

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

$$±51,±17,±3,±1$$

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.

$$x=-3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+x^{2}+11x+51 by x+3 to get x^{2}-2x+17. Solve the equation where the result equals to 0.

$$x^{2}-2x+17=0$$

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 17 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\times 17}}{2}$$

Do the calculations.

$$x=\frac{2±\sqrt{-64}}{2}$$

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.

$$x=-3$$