Solve x^3-8x^2+15x

$$x ^ { 3 } - 8 x ^ { 2 } + 15 x$$

Factor

$$x\left(x-5\right)\left(x-3\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}-8x+15\right)$$

Consider x^{2}-8x+15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\times 15=15$$

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 15.

$$-1,-15 -3,-5$$

Calculate the sum for each pair.

$$-1-15=-16 -3-5=-8$$

The solution is the pair that gives sum -8.

$$a=-5 b=-3$$

Rewrite x^{2}-8x+15 as \left(x^{2}-5x\right)+\left(-3x+15\right).

$$\left(x^{2}-5x\right)+\left(-3x+15\right)$$

Factor out x in the first and -3 in the second group.

$$x\left(x-5\right)-3\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(x-3\right)$$

Rewrite the complete factored expression.

$$x\left(x-5\right)\left(x-3\right)$$

Evaluate

$$x\left(x-5\right)\left(x-3\right)$$