Form a polynomial f(x) with real coefficients having the degree and zeros

Question

degree: 4 ; zeros:1,2 and 1-2i

Answer 1

degree: 4 (tells us that we need four zeros)

zeros:1,2 (tells us (x-1) and (x-2) are two factors of f(x))

and

1-2i (implies that the complex conjugate 1+2i is another zero)

 

f(x) = (x-1)(x-2)(some quadratic whose roots are 1±2i)

 

Let's find that quadratic by finding SUM and PRODUCT of the complex roots:

S = 1 + 2i + 1 - 2i = 2

P = (1 + 2i)(1 - 2i) = 1-4i² = 1+4 = 5

 

Substituting into quadratic equation form x² - Sx + P = 0:

x² - 2x + 5 = 0

 

Last, put this quadratic with complex roots into our f(x):

 

f(x) = (x-1)(x-2)(some quadratic whose roots are 1±2i)

f(x) = (x-1)(x-2)(x² - 2x + 5)   (you may expand it further if you'd like!)

 

Hope this helps!