Describe the number and type of roots for the polynomial (how many real and complex?) x^3+5x^2-4x-2=0

Question

Describe the number and type of roots for the polynomial (how many real and complex?) x^3+5x^2-4x-2=0

Answer 1

First, we use synthetic division to find a quadratic factor.

 

1 | 1     5    -4      -2

 

            1     6       2

__________________

     1     6     2        0

 

 

Then the factored form of the left side is

 

(x - 1)(x² + 6x + 2) = 0

 

 

Next, use the discriminant of the quadratic factor to see if it is positive, negative or zero.  If negative, then complex roots.  Otherwise, real roots.

 

Discriminant = 6² - 4(1)(2)

                   = 36 - 8

                   = 28

 

Therefore, you have 3 real roots. 1 rational root and 2 irrational roots.  You have 2 irrational roots because 28 is not a perfect square.