Polynomial function of least degree having only real coefficients

Question

Has a leading coefficient of 1, and zeros of 5 and 3+i

Answer 1

In order to have real coefficients, the complex conjugate of a complex zero must also be a zero.  Therefore the zeros are

 

5

3 + i

3 - i

 

f(x) = A(x - 5) [x - (3 + i)] [x - (3 - i)]

 

A = leading coefficient = 1

 

f(x) = (x - 5) [x - (3 + i)] [x - (3 - i)]

 

Expand this into standard from very carefully.  The coefficients should all come out real.