Solve x^4-2*x^3/3-31*x^2/3+8*x/3+2=0

$$x ^ { 4 } - \frac { 2 \cdot x ^ { 3 } } { 3 } - \frac { 31 \cdot x ^ { 2 } } { 3 } + \frac { 8 \cdot x } { 3 } + 2 = 0$$

Solve for x

$$x=2-\sqrt{2}\approx 0.585786438$$

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

$$x=\sqrt{2}+2\approx 3.414213562$$

$$x=-3$$

Solution Steps

Multiply both sides of the equation by 3.

$$3x^{4}-2x^{3}-31x^{2}+8x+6=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±2,±6,±1,±3,±\frac{2}{3},±\frac{1}{3}$$

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.

$$x=-3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{4}-2x^{3}-31x^{2}+8x+6 by x+3 to get 3x^{3}-11x^{2}+2x+2. Solve the equation where the result equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

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

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}-11x^{2}+2x+2 by 3\left(x+\frac{1}{3}\right)=3x+1 to get x^{2}-4x+2. Solve the equation where the result equals to 0.

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

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 2 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{4±2\sqrt{2}}{2}$$

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

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

List all found solutions.

$$x=-3 x=-\frac{1}{3} x=2-\sqrt{2} x=\sqrt{2}+2$$