Solve 3x^3+7x^2-22x-8=0

$$3 x ^ { 3 } + 7 x ^ { 2 } - 22 x - 8 = 0$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x=2$$

$$x=-4$$

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 -8 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{8}{3},±8,±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}+7x^{2}-22x-8 by x-2 to get 3x^{2}+13x+4. Solve the equation where the result equals to 0.

$$3x^{2}+13x+4=0$$

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 3 for a, 13 for b, and 4 for c in the quadratic formula.

$$x=\frac{-13±\sqrt{13^{2}-4\times 3\times 4}}{2\times 3}$$

Do the calculations.

$$x=\frac{-13±11}{6}$$

Solve the equation 3x^{2}+13x+4=0 when ± is plus and when ± is minus.

$$x=-4 x=-\frac{1}{3}$$

List all found solutions.

$$x=2 x=-4 x=-\frac{1}{3}$$