Factor remainder theorem

Question

Given that (x+2) is a factor of x³ + 5x² + ax + 2, find the value of a.

Answer 1

        x² + 3x + 1

      ----------------------

x+2|x³ + 5x² + ax + 2

     -x³ -  2x²

     ------------

              3x² + ax

             -3x² - 6x

            -----------

                    ax-6x+2

                         -x -2

                       -----------

                              0

 

We need to have x = 1 in this last bit:

a - 6 = 1

a = 7

 

Checking:

(x + 2)(x² + 3x + 1)

x³ + 3x² + x + 2x² + 6x + 2

Combine like terms:

x³ + 5x² + 7x + 2, which is our original expression.

 

so 'a' must be 7.