Solve x^3+16x^2+28x

$${ x }^{ 3 } +16 { x }^{ 2 } +28x$$

Factor

$$x\left(x+2\right)\left(x+14\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}+16x+28\right)$$

Consider x^{2}+16x+28. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\times 28=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 16.

$$a=2 b=14$$

Rewrite x^{2}+16x+28 as \left(x^{2}+2x\right)+\left(14x+28\right).

$$\left(x^{2}+2x\right)+\left(14x+28\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x+2\right)+14\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+14\right)$$

Rewrite the complete factored expression.

$$x\left(x+2\right)\left(x+14\right)$$

Evaluate

$$x\left(x+2\right)\left(x+14\right)$$