Thread: Polynomial with real coefficients

I have a problem that I am not understanding for homework:

Find a polynomial with real coefficients, degree of 3, zeros of 2,1,-i, and satisfies the condition that f(0)=4

I assumed that thus the constant would be '4'... (ax^3 + ax^2 +ax +4)
And I assumed that the zeros would meand: (x-2)(x-1) and (x^2+1)

The problem that I am facing is that the above would result in a degree of 4, not 3. Is there some small bit I am missing, or am I misinterperting (sp?) the question.

Hi, airyie!

Originally Posted by airyie

The problem that I am facing is that the above would result in a degree of 4, not 3. Is there some small bit I am missing, or am I misinterperting (sp?) the question.

This problem has no solution. If is a root of a polynomial with real coefficients in one variable, then is also a root of that polynomial. This implies that, for your polynomial to have a root of , it must also have a root of . And, since the fundamental theorem of algebra states that a polynomial with complex coefficients of degree can have no more than complex roots, it follows that the polynomial sought in the problem must be of degree 4 or higher.

Now, if you want to find a fourth-degree polynomial with the given characteristics, you have the right idea: