Find a degree 3 polynomial with real coefficients having zeros 5 and 5 - 5i and a lead coefficient of 1. Write P in expanded form.

Question

Find a degree 3 polynomial with real coefficients having zeros
5 and 5 - 5i and a lead coefficient of 1. Write P in expanded form. 

Answer 1

P(x) = (x - 5)(x - (5 - 5i))(x - (5 + 5i))

 

You get to expand.

Answer 2

A zero at 5 means (x-5) is a factor

A zero at 5-5i means (x - (5-5i)) or (x-5+5i) is a factor

If 5-5i is a factor then so is the complex conjugate 5+5i:

A zero at 5+5i means (x-(5+5i)) or (x-5-5i) is a factor

 

p(x) = a(x-5)(x-5+5i)(x-5-5i)

Since the leading coefficient is 1, a=1

 

p(x) = (x-5)(x-5+5i)(x-5-5i)

 

p(x) = (x² - 5x + 5ix - 5x + 25 - 25i)(x-5-5i)

p(x) = (x² - 10x + 25 + 5ix - 25i)(x-5-5i)

 

multiplying through by each term and adding like terms gives us:

 

p(x) =  x³ - 5x²- 5ix²

              - 10x²         + 50x + 50ix

                                + 25x             - 125 - 125i

                      +5ix²  + 25x  - 25ix

                                          - 25ix    - 125  +125i

     -----------------------------------------------------------

p(x) = x³ - 15x² + 100x - 250