Solve 12t^3-10t^2-12t

$$12 t ^ { 3 } - 10 t ^ { 2 } - 12 t$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(6t^{3}-5t^{2}-6t\right)$$

Consider 6t^{3}-5t^{2}-6t. Factor out t.

$$t\left(6t^{2}-5t-6\right)$$

Consider 6t^{2}-5t-6. Factor the expression by grouping. First, the expression needs to be rewritten as 6t^{2}+at+bt-6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-6\right)=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -5.

$$a=-9 b=4$$

Rewrite 6t^{2}-5t-6 as \left(6t^{2}-9t\right)+\left(4t-6\right).

$$\left(6t^{2}-9t\right)+\left(4t-6\right)$$

Factor out 3t in the first and 2 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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