Solve 3t+2t^2-t^3

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

Factor

$$-t\left(t-3\right)\left(t+1\right)$$

Solution Steps

Factor out t.

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

Consider 3+2t-t^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as -t^{2}+at+bt+3. To find a and b, set up a system to be solved.

$$a+b=2 ab=-3=-3$$

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.

$$a=3 b=-1$$

Rewrite -t^{2}+2t+3 as \left(-t^{2}+3t\right)+\left(-t+3\right).

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

Factor out -t in the first and -1 in the second group.

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

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

$$\left(t-3\right)\left(-t-1\right)$$

Rewrite the complete factored expression.

$$t\left(t-3\right)\left(-t-1\right)$$

Evaluate

$$-t\left(t-3\right)\left(t+1\right)$$