How do I use the conjugate zeros theorem?

1 Answer
Jim H
Apr 26, 2015

As a student, if a teacher tells you that a polynomial with real coefficients has 3i for one of its zeros, that you can reason:

With real corrifients, if 3i is a zero, then so is -3i. So I know that (x-3i) and (x-(-3i)) = (x+3i) are both factors.

So I know that (x-3i)(x+3i) = x^2 + 9 is a factor. That might help me factor the polynomial. (If I've learned division of polynomials.)

I saw this interesting bit of reasoning recently here on Socratic.

We know that a polynomial of degree n has n zeros (counting multiplicity). So a polynomial of odd degree has an odd number of zeros.
We also know that if a polynomial has real coefficients, the any imaginary zeros occur in conjugate pairs.

So we can conclude that a polynomial with real coefficients and odd degee must have at least one real zero. (An odd number of zeros cannot all be imaginary.)

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