Solve 3x^3-13x^2-9x+39=0

$$3 x ^ { 3 } - 13 x ^ { 2 } - 9 x + 39 = 0$$

Solve for x

$$x=-\sqrt{3}\approx -1.732050808$$

$$x = \frac{13}{3} = 4\frac{1}{3} \approx 4.333333333$$

$$x=\sqrt{3}\approx 1.732050808$$

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

$$±13,±39,±\frac{13}{3},±1,±3,±\frac{1}{3}$$

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.

$$x=\frac{13}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}-13x^{2}-9x+39 by 3\left(x-\frac{13}{3}\right)=3x-13 to get x^{2}-3. Solve the equation where the result equals to 0.

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

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, 0 for b, and -3 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-3\right)}}{2}$$

Do the calculations.

$$x=\frac{0±2\sqrt{3}}{2}$$

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

$$x=-\sqrt{3} x=\sqrt{3}$$

List all found solutions.

$$x=\frac{13}{3} x=-\sqrt{3} x=\sqrt{3}$$