solve an equation if -2 is a root

Question

 

 

how do I solve the equation x^3 + 2x^2 + x + 2 =0, if -2 is a root

 

 

thanks

Answer 1

How do I solve the equation x^3 + 2x^2 + x + 2 =0, if -2 is a root

 

You can divide the polynomial by (x+2):

 

           _x²___0___1___

(x+2) ) x³ + 2x² + x + 2

         -(x³ + 2x²)

          -----------

            0      0      x + 2  

                         -(x + 2)

                          ---------

                            0    0

 

The quotient is x² + 1.  Since there is no remainder, (x+2) is a solution.  Using the quotient from the division, the factors so far are:

 

(x+2)(x²+1)=0

 

x²+1 = 0 when x = ± i.  Hence the roots are:

 

x = -2, +i, -i

 

Check:

 

(-2)³ + 2(-2)² + (-2) + 2 = -8 + 8 - 2 + 2 = 0

 

(i)³ + 2(i)² + i + 2 = -i -2 + i + 2 = 0       

 

(-i)³ + 2(-i)² - i + 2 = i - 2 - i + 2 = 0