Solve f(x)=x^3-5x^2+4x

$$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 4 x$$

Factor

$$x\left(x-4\right)\left(x-1\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}-5x+4\right)$$

Consider x^{2}-5x+4. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

Rewrite x^{2}-5x+4 as \left(x^{2}-4x\right)+\left(-x+4\right).

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

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

$$x\left(x-4\right)-\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$x\left(x-4\right)\left(x-1\right)$$

Evaluate

$$x\left(x-4\right)\left(x-1\right)$$