Solve g(x)=x^3-6x^2+8x

$$g ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x$$

Factor

$$x\left(x-4\right)\left(x-2\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}-6x+8\right)$$

Consider x^{2}-6x+8. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\times 8=8$$

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 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -6.

$$a=-4 b=-2$$

Rewrite x^{2}-6x+8 as \left(x^{2}-4x\right)+\left(-2x+8\right).

$$\left(x^{2}-4x\right)+\left(-2x+8\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(x-4\right)-2\left(x-4\right)$$

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

$$\left(x-4\right)\left(x-2\right)$$

Rewrite the complete factored expression.

$$x\left(x-4\right)\left(x-2\right)$$

Evaluate

$$x\left(x-4\right)\left(x-2\right)$$