Solve x^3+5x^2-24x

$$x ^ { 3 } + 5 x ^ { 2 } - 24 x$$

Factor

$$x\left(x-3\right)\left(x+8\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}+5x-24\right)$$

Consider x^{2}+5x-24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=-3 b=8$$

Rewrite x^{2}+5x-24 as \left(x^{2}-3x\right)+\left(8x-24\right).

$$\left(x^{2}-3x\right)+\left(8x-24\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x-3\right)+8\left(x-3\right)$$

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

$$\left(x-3\right)\left(x+8\right)$$

Rewrite the complete factored expression.

$$x\left(x-3\right)\left(x+8\right)$$

Evaluate

$$x\left(x-3\right)\left(x+8\right)$$