Solve (m^3-m^2+100m-100)=0

$$( m ^ { 3 } - m ^ { 2 } + 100 m - 100 ) = 0$$

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 -100 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±100,±50,±25,±20,±10,±5,±4,±2,±1$$

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.

$$m=1$$

By Factor theorem, m-k is a factor of the polynomial for each root k. Divide m^{3}-m^{2}+100m-100 by m-1 to get m^{2}+100. Solve the equation where the result equals to 0.

$$m^{2}+100=0$$

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 100 for c in the quadratic formula.

$$m=\frac{0±\sqrt{0^{2}-4\times 1\times 100}}{2}$$

Do the calculations.

$$m=\frac{0±\sqrt{-400}}{2}$$

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.

$$m=1$$

Solve for m (complex solution)

$$m=-10i$$

$$m=1$$

$$m=10i$$