Zeros of Polynomial Functions

Question

Find a polynomial function f(x) of least degree having real coefficients with zeros as given.

 

7 - 2i and 7 + 2i

Answer 1

f(x) = [x-(7-2i)][x-(7+2i)]

 

      = [(x-7)+2i][(x-7)-2i]

 

      = (x-7)² - (2i)²

 

      = x² - 14x + 49 - 4i²  = x² - 14x + 49 +4

 

                                    = x² - 14x + 53

 

 

Answer 2

To supplement the previous answers, one reason degree two is least because a degree one polynomial with real coefficients has the form ax+b, hence has only real roots (and only one).

 

Also, technically, both those complex numbers given are roots of the zero polynomial (do look up the definition of zero of a function), which clearly has least degree. f(x) = 0

Answer 3

f(x) = (x - (7 - 2i))(7 - (7 + 2i))     factor/root relation

 

Can you multiply this out?