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

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

Factor

$$x\left(x-1\right)\left(x+7\right)$$

Solution Steps

Factor out x.

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

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

$$a+b=6 ab=1\left(-7\right)=-7$$

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. The only such pair is the system solution.

$$a=-1 b=7$$

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

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

Factor out x in the first and 7 in the second group.

$$x\left(x-1\right)+7\left(x-1\right)$$

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

$$\left(x-1\right)\left(x+7\right)$$

Rewrite the complete factored expression.

$$x\left(x-1\right)\left(x+7\right)$$

Evaluate

$$x\left(x-1\right)\left(x+7\right)$$