Solve x^4-4x^3+3x^2

$${ x }^{ 4 } -4 { x }^{ 3 } +3 { x }^{ 2 }$$

Factor

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

Solution Steps

Factor out x^{2}.

$$x^{2}\left(x^{2}-4x+3\right)$$

Consider x^{2}-4x+3. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\times 3=3$$

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.

$$a=-3 b=-1$$

Rewrite x^{2}-4x+3 as \left(x^{2}-3x\right)+\left(-x+3\right).

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

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

$$x\left(x-3\right)-\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x-1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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