INTRODUCTORY EXERCISES

(8) Use the fact about roots of polynomials (the Factor Theorem) to find all real roots of the following polynomial equation.

   a3 - 6 = 6a2 - 11a 

   a3 - 6a2 + 11a - 6 = 0

Using long division,

             a2 - 4a + 3
           |-------------------
    (a - 2)|a3 - 6a2 + 11a - 6
          -(a3 - 2a2)
            -------------
                -4a2 + 11a - 6
              -(-4a2 +  8a)
                ---------------
                        3a - 6
                      -(3a - 6)
                        ------
                           0

Therefore, a3 - 6a2 + 6a - 6 =
           (a - 2)(a2 - 4a + 3) =(a - 2)(a - 3)(a - 1) 

The roots of a3 - 6 = 6a2 - 11a are: 1, 2, and 3.