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Math Help - Polynomial with real coefficients

  1. #1
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    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.
    Last edited by airyie; May 19th 2008 at 06:02 PM. Reason: efficiency
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    Hi, airyie!

    Quote Originally Posted by airyie View Post
    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 a,\;b\in\mathbb{R}\text{ and }a+b\,\text{i} is a root of a polynomial with real coefficients in one variable, then a-b\,\text{i} is also a root of that polynomial. This implies that, for your polynomial to have a root of -\text{i}, it must also have a root of +\text{i}. And, since the fundamental theorem of algebra states that a polynomial with complex coefficients of degree n can have no more than n 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:

    f(x) = a\left(x-2\right)\left(x-1\right)\left(x^2+1\right)

    f(0) = 2a = 4 \Rightarrow a = 2

    \Rightarrow f(x) = 2\left(x-2\right)\left(x-1\right)\left(x^2+1\right) = 2x^4-6x^3+6x^2-6x+4
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