Solve m^4-7m^3+18m^2-20m+8

$$m ^ { 4 } - 7 m ^ { 3 } + 18 m ^ { 2 } - 20 m + 8$$

Factor

$$\left(m-1\right)\left(m-2\right)^{3}$$

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 8 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by m-2.

$$\left(m-2\right)\left(m^{3}-5m^{2}+8m-4\right)$$

Consider m^{3}-5m^{2}+8m-4. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. One such root is 2. Factor the polynomial by dividing it by m-2.

$$\left(m-2\right)\left(m^{2}-3m+2\right)$$

Consider m^{2}-3m+2. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite m^{2}-3m+2 as \left(m^{2}-2m\right)+\left(-m+2\right).

$$\left(m^{2}-2m\right)+\left(-m+2\right)$$

Factor out m in the first and -1 in the second group.

$$m\left(m-2\right)-\left(m-2\right)$$

Factor out common term m-2 by using distributive property.

$$\left(m-2\right)\left(m-1\right)$$

Rewrite the complete factored expression.

$$\left(m-1\right)\left(m-2\right)^{3}$$

Evaluate

$$\left(m-1\right)\left(m-2\right)^{3}$$