Complex Conjugate Zeros

Key Questions

  • How do I find a complex conjugate?

    The complex conjugate of a complex number a+bi can be found by negating the imaginary part, that is, a-bi.

    For example, the complex conjugate of 2-3i is 2+3i.

    Wataru · · Sep 26 2014
  • What is the conjugate zeros theorem?

    Answer:

    If a polynomial has Real coefficients, then any Complex zeros will occur in Complex conjugate pairs.

    That is, if z = a+bi is a zero then bar(z) = a-bi is also a zero.

    Explanation:

    Actually a similar theorem holds for square roots and polynomials with rational coefficients:

    If f(x) is a polynomial with rational coefficients and a zero expressible in the form a+b sqrt(c) where a, b, c are rational and sqrt(c) is irrational, then a-b sqrt(c) is also a zero.

    George C. · 1 · Nov 16 2015
  • How do I use the conjugate zeros theorem?

    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.)

    Jim H · 1 · Apr 26 2015
  • What is a complex conjugate?

    A complex conjugate is the number to which you have to multiply a complex number in order to make it real.
    By using the identity (x+y) . (x-y) = x²-y², we see that, to every complex, there is another to which we can multiply it in order to get a new number that will not depend on i.
    If (a+bi).(c+di) is real, (c+di) is (a+bi)'s conjugate and it equals (a-bi).

    Bernardo Russo G. Ferreira · · Oct 3 2014

Questions

Complex Zeros