Solve 5x^3+55x^2+120x

$$5 x ^ { 3 } + 55 x ^ { 2 } + 120 x$$

Factor

$$5x\left(x+3\right)\left(x+8\right)$$

Solution Steps

Factor out 5.

$$5\left(x^{3}+11x^{2}+24x\right)$$

Consider x^{3}+11x^{2}+24x. Factor out x.

$$x\left(x^{2}+11x+24\right)$$

Consider x^{2}+11x+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.

$$a+b=11 ab=1\times 24=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

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

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 11.

$$a=3 b=8$$

Rewrite x^{2}+11x+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.

$$5x\left(x+3\right)\left(x+8\right)$$

Evaluate

$$5x\left(x+3\right)\left(x+8\right)$$