Solve x^3-2x^2-15x

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

Factor

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

Solution Steps

Factor out x.

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

Consider x^{2}-2x-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.

$$a+b=-2 ab=1\left(-15\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -15.

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

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -2.

$$a=-5 b=3$$

Rewrite x^{2}-2x-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

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