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Math Help - find a real solution for a complex root

  1. #1
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    find a real solution for a complex root

    i am having problems understanding the roots
    have the equation:
    given that x = 4 is a real solution for f(x) = x^3 - 4x^2 + x - 4, find one of the complex roots?

    and the steps to find a cube root of -8?
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    Re: find a real solution for a complex root

    Quote Originally Posted by tman View Post
    i am having problems understanding the roots
    have the equation:
    given that x = 4 is a real solution for f(x) = x^3 - 4x^2 + x - 4, find one of the complex roots?

    and the steps to find a cube root of -8?
    Since x = 4 is a solution, that means (x - 4) is a factor. Long divide to get the quadratic factor, and factorise that over the complex numbers to evaluate the other two solutions (they will be complex conjugates).

    To find \displaystyle \begin{align*} \sqrt[3]{-8} \end{align*}...

    \displaystyle \begin{align*} z^3 &= -8 \\ z^3 + 8 &= 0 \\ (z + 2)\left( z^2 - 2z + 4 \right) &= 0 \\ z + 2 = 0 \textrm{ or } z^2 - 2z + 4 &= 0 \end{align*}

    Solve each of those two equations to find the three cube roots of -8.
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    Re: find a real solution for a complex root

    an only slighter easier way:

    suppose we already know how to find the 3 cube roots of 1 (by, say, using trigonometry), call these: 1, ω, and ω2 (= \overline{\omega}).

    (it should be easy to see that if ω3 = 1, and ω ≠ 1, then (ω2)3 = (ω3)2 = (1)(1) = 1, as well).

    (we can find these by factoring x3 - 1 = (x - 1)(x2 + x + 1), also).

    then (x + 2)(x + 2ω)(x + 2ω2) = (x + 2)(x2 + (2ω + 2ω2)x + 4ω3)

    = (x + 2)(x2 + 2(ω + ω2)x + 4), and we know from above that: ω + ω2 = -1, so:

    = (x + 2)(x2 - 2x2 + 4) = x3 + 8

    so the three cube roots of -8 are -2,-2ω, and -2ω2.
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    Re: find a real solution for a complex root

    you mean to factor out a x to get x^2-4x+4
    then (x-2)(x-2)/x-4?
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    Re: find a real solution for a complex root

    Quote Originally Posted by tman View Post
    you mean to factor out a x to get x^2-4x+4
    then (x-2)(x-2)/x-4?
    I don't know who you're speaking to, but no, there is no factoring of a single x. You are going to long divide \displaystyle \begin{align*} \frac{x^3 - 4x^2 + x - 4}{x - 4} \end{align*}.

    The top is also able to be factorised by grouping...

    \displaystyle \begin{align*} x^3 - 4x^2 + x - 4 &= x^3 + x - 4x^2 - 4 \\ &= x\left( x^2 + 1 \right) - 4 \left( x^2 + 1 \right) \\ &= (x - 4) \left( x^2 + 1 \right) \\ &= (x - 4) (x - i)(x + i) \end{align*}
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    Re: find a real solution for a complex root

    Quote Originally Posted by Deveno View Post
    an only slighter easier way:

    suppose we already know how to find the 3 cube roots of 1 (by, say, using trigonometry), call these: 1, ω, and ω2 (= \overline{\omega}).

    (it should be easy to see that if ω3 = 1, and ω ≠ 1, then (ω2)3 = (ω3)2 = (1)(1) = 1, as well).

    (we can find these by factoring x3 - 1 = (x - 1)(x2 + x + 1), also).

    then (x + 2)(x + 2ω)(x + 2ω2) = (x + 2)(x2 + (2ω + 2ω2)x + 4ω3)

    = (x + 2)(x2 + 2(ω + ω2)x + 4), and we know from above that: ω + ω2 = -1, so:

    = (x + 2)(x2 - 2x2 + 4) = x3 + 8

    so the three cube roots of -8 are -2,-2ω, and -2ω2.
    I wouldn't say that is easier, it seems like more work than the method I gave. Each to their own
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