Solve a^3-6a^2+11a-6

$$a ^ { 3 } - 6 a ^ { 2 } + 11 a - 6$$

Factor

$$\left(a-3\right)\left(a-2\right)\left(a-1\right)$$

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 a-3.

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

Consider a^{2}-3a+2. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+2. To find p and q, set up a system to be solved.

$$p+q=-3 pq=1\times 2=2$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. The only such pair is the system solution.

$$p=-2 q=-1$$

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

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

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

$$a\left(a-2\right)-\left(a-2\right)$$

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

$$\left(a-2\right)\left(a-1\right)$$

Rewrite the complete factored expression.

$$\left(a-3\right)\left(a-2\right)\left(a-1\right)$$

Evaluate

$$\left(a-3\right)\left(a-2\right)\left(a-1\right)$$