second degree polynomials ?

Question

The question asks:

 

Write a second degree polynimial equation with real coefficients and the given root.

 

4+2i

 

Answer 1

Recall that if z is a root of a real polynomial p, then so is the conjugate z*. You're given 4 + 2i is a root, so the conjugate 4 - 2i is also a root. Since your real polynomial must be second degree, these are the only roots. So consider 

 

p(x) = (x - (4 + 2i))(x - (4 - 2i)) = ((x - 4) - 2i)((x - 4) + 2i)

 

and use the formula (a - b)(a + b) = a² - b² with a = x - 4 and b = 2i to obtain 

 

p(x) = (x - 4)^2 - (2i)^2 = (x - 4)^2 - (-4) = (x - 4)^2 + 4.

 

Now expand (x - 4)^2 = x^2 - 8x + 16 to get 

 

p(x) = x^2 - 8x + 20.

 

The polynomial p satisfies all the desired conditions.