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Complex numbers help ??

  • 22-01-2012 11:30pm
    #1
    Irish Fox
    Closed Accounts Posts: 41 ✭


    Need help with these questions

    f(z) = z^2 + 6z + 25 is a quadratic polynomial

    verify that z = - 3 - 4i is a root of f(z)

    Write down the other root of the equation

    z^2 + 6z + 25 = 0

    _____________

    f(z) = z^3 - z^2 - 4z - 6 is a cubic polynomial

    show that z = 1 + i is a solution to the equation f(z) = 0


Comments

  • #2
    MathsManiac
    Registered Users, Registered Users 2 Posts: 1,595 ✭✭✭MathsManiac


    To verify that a number is a root of a quadratic or cubic equation, you substitute it in for the variable and show that it works.

    So, for the first one, you need to show, by working it out, that
    (-3-4i)^2 + 6(-3-4i) +25 = 0.

    For the follow-on bit, you use a theorem called the "conjugate roots theorem". This says that in quadratic or cubic where the coefficients are real numbers, then the conjugate of a root is also a root. That is, if one root is a+bi, then a-bi is also a root.

    Your second question is similar to the first: substitute and work it out to verify.


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