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

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

Factor

$$x\left(3x-2\right)\left(x+3\right)$$

Solution Steps

Factor out x.

$$x\left(3x^{2}+7x-6\right)$$

Consider 3x^{2}+7x-6. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=7 ab=3\left(-6\right)=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 7.

$$a=-2 b=9$$

Rewrite 3x^{2}+7x-6 as \left(3x^{2}-2x\right)+\left(9x-6\right).

$$\left(3x^{2}-2x\right)+\left(9x-6\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(3x-2\right)+3\left(3x-2\right)$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$x\left(3x-2\right)\left(x+3\right)$$

Evaluate

$$x\left(3x-2\right)\left(x+3\right)$$