Sample Problem

Which of the following third-degree polynomial functions (with real coefficients)

has zeros at x=5+i and x=1?


Because we know that 5+i is a zero and the polynomial has to have real coefficients,

we know that the conjugate 5-i must also be a zero. So from the Linear Factorization

Theorem, f(x) can be written as

f(x)=a(x-1)(x-(5+i))(x-(5-i)); let’s take a=1 for simplicity’s sake

f(x)=(x-1)(x-(5+i))(x-(5-i))=(x-1)(x-5-i)(x-5+i)=(x-1)(x2-10x+26)= x3-11x2+36x-26