Solve t^3-2t^2-5t+6

$$t ^ { 3 } - 2 t ^ { 2 } - 5 t + 6$$

Factor

$$\left(t-3\right)\left(t-1\right)\left(t+2\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 1. One such root is 3. Factor the polynomial by dividing it by t-3.

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

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 positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=2$$

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

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

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

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

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

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

Rewrite the complete factored expression.

$$\left(t-3\right)\left(t-1\right)\left(t+2\right)$$

Evaluate

$$\left(t-3\right)\left(t-1\right)\left(t+2\right)$$