Complex Conjugate Zeros

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Key Questions

  • 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#).

  • 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#.

  • 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.

  • 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.)

Questions

  • mason m answered · 1 year ago