Solve x^3-3x^2-68x+280=0

$${ x }^{ 3 } -3 { x }^{ 2 } -68x+280=0$$

Solve for x

$$x=2\sqrt{11}-2\approx 4.633249581$$

$$x=-2\sqrt{11}-2\approx -8.633249581$$

$$x=7$$

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

$$±280,±140,±70,±56,±40,±35,±28,±20,±14,±10,±8,±7,±5,±4,±2,±1$$

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.

$$x=7$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-3x^{2}-68x+280 by x-7 to get x^{2}+4x-40. Solve the equation where the result equals to 0.

$$x^{2}+4x-40=0$$

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, 4 for b, and -40 for c in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 1\left(-40\right)}}{2}$$

Do the calculations.

$$x=\frac{-4±4\sqrt{11}}{2}$$

Solve the equation x^{2}+4x-40=0 when ± is plus and when ± is minus.

$$x=-2\sqrt{11}-2 x=2\sqrt{11}-2$$

List all found solutions.

$$x=7 x=-2\sqrt{11}-2 x=2\sqrt{11}-2$$