Solve x^3+7x^2+6x

$$x ^ { 3 } + 7 x ^ { 2 } + 6 x$$

Factor

$$x\left(x+1\right)\left(x+6\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}+7x+6\right)$$

Consider x^{2}+7x+6. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\times 6=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 7.

$$a=1 b=6$$

Rewrite x^{2}+7x+6 as \left(x^{2}+x\right)+\left(6x+6\right).

$$\left(x^{2}+x\right)+\left(6x+6\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x+1\right)+6\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$x\left(x+1\right)\left(x+6\right)$$

Evaluate

$$x\left(x+1\right)\left(x+6\right)$$