Solve 2m^3-m^2+m-2=0

$$2 m ^ { 3 } - m ^ { 2 } + m - 2 = 0$$

Solve for m

$$m=1$$

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 -2 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±1,±2,±\frac{1}{2}$$

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.

$$m=1$$

By Factor theorem, m-k is a factor of the polynomial for each root k. Divide 2m^{3}-m^{2}+m-2 by m-1 to get 2m^{2}+m+2. Solve the equation where the result equals to 0.

$$2m^{2}+m+2=0$$

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 2 for a, 1 for b, and 2 for c in the quadratic formula.

$$m=\frac{-1±\sqrt{1^{2}-4\times 2\times 2}}{2\times 2}$$

Do the calculations.

$$m=\frac{-1±\sqrt{-15}}{4}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$m\in \emptyset$$

List all found solutions.

$$m=1$$