Solve x^3-x^2-17x-15=0

$$x ^ { 3 } - x ^ { 2 } - 17 x - 15 = 0$$

Solve for x

$$x=-1$$

$$x=5$$

$$x=-3$$

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 -15 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±15,±5,±3,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-x^{2}-17x-15 by x+1 to get x^{2}-2x-15. Solve the equation where the result equals to 0.

$$x^{2}-2x-15=0$$

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 -15 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\left(-15\right)}}{2}$$

Do the calculations.

$$x=\frac{2±8}{2}$$

Solve the equation x^{2}-2x-15=0 when ± is plus and when ± is minus.

$$x=-3 x=5$$

List all found solutions.

$$x=-1 x=-3 x=5$$