Solve 3x^3-22x^2+37x-10=0

$$3 x ^ { 3 } - 22 x ^ { 2 } + 37 x - 10 = 0$$

Solve for x

$$x=2$$

$$x=5$$

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

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 -10 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{10}{3},±10,±\frac{5}{3},±5,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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}-22x^{2}+37x-10 by x-2 to get 3x^{2}-16x+5. Solve the equation where the result equals to 0.

$$3x^{2}-16x+5=0$$

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, -16 for b, and 5 for c in the quadratic formula.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 3\times 5}}{2\times 3}$$

Do the calculations.

$$x=\frac{16±14}{6}$$

Solve the equation 3x^{2}-16x+5=0 when ± is plus and when ± is minus.

$$x=\frac{1}{3} x=5$$

List all found solutions.

$$x=2 x=\frac{1}{3} x=5$$