Solve t^3-t^2-2t

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

Factor

$$t\left(t-2\right)\left(t+1\right)$$

Solution Steps

Factor out t.

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

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

$$a+b=-1 ab=1\left(-2\right)=-2$$

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. The only such pair is the system solution.

$$a=-2 b=1$$

Rewrite t^{2}-t-2 as \left(t^{2}-2t\right)+\left(t-2\right).

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

Factor out t in t^{2}-2t.

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

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

$$\left(t-2\right)\left(t+1\right)$$

Rewrite the complete factored expression.

$$t\left(t-2\right)\left(t+1\right)$$

Evaluate

$$t\left(t-2\right)\left(t+1\right)$$