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Math Help - Need Help on a roots question please!

  1. #1
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    Need Help on a roots question please!

    If x=i is one root of x^3 + x^2 + x + 1, find the other two roots.

    Thank you!
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  2. #2
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    Quote Originally Posted by antz215 View Post
    If x=i is one root of x^3 + x^2 + x + 1, find the other two roots.

    Thank you!

    Well the complex roots of a real polynomial occur in conjugate pairs, so if x=i is a root so is x=-i, so x^2+1 is a factor of your cubic.

    Now divide x^3+x^2+x+1 by x^2+1 to get the remaining linear factor and hence find the final root.

    RonL
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  3. #3
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    thats what i had thought, but both people i asked had no idea haha. thank you so much!
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  4. #4
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    I don't get this. If x=i and x=-i, that would mean i=-i, which I'm pretty sure is not true. It's been a while since pre-calc for me. What is this root lingo that allows both x=i and x=-i ?
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by hatsoff View Post
    I don't get this. If x=i and x=-i, that would mean i=-i, which I'm pretty sure is not true. It's been a while since pre-calc for me. What is this root lingo that allows both x=i and x=-i ?
    no.

    both i and -i being roots does not imply they are equal.

    look at x^2 - 1 = 0

    the roots are 1 and -1, does that mean 1 = -1? of course not

    CaptainBlack called both x, but he was suggesting that these are the x-values that give roots for the equation, not that they are equal. an abuse of notation, i know, but it is understood in the context

    complex roots come in conjugate pairs. we get complex roots when we take even roots of negative numbers, these roots therefore come in +/- form
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  6. #6
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    Quote Originally Posted by hatsoff View Post
    I don't get this. If x=i and x=-i, that would mean i=-i, which I'm pretty sure is not true. It's been a while since pre-calc for me. What is this root lingo that allows both x=i and x=-i ?
    It's your English/Maths comprehension that is lacking here.

    What I wrote is: "If x=i is a root, then so is x=-i", that is when x=i the polynomial is equal to 0, and when x=-1 the polynomial is also equal to 0.

    RonL
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    It's your English/Maths comprehension that is lacking here.

    What I wrote is: "If x=i is a root, then so is x=-i", that is when x=i the polynomial is equal to 0, and when x=-1 the polynomial is also equal to 0.

    RonL
    Gotcha. I feel silly, now.

    Thanks!
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  8. #8
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    Quote Originally Posted by Jhevon View Post
    no.

    both i and -i being roots does not imply they are equal.

    look at x^2 - 1 = 0

    the roots are 1 and -1, does that mean 1 = -1? of course not

    CaptainBlack called both x, but he was suggesting that these are the x-values that give roots for the equation, not that they are equal. an abuse of notation, i know, but it is understood in the context

    complex roots come in conjugate pairs. we get complex roots when we take even roots of negative numbers, these roots therefore come in +/- form
    Thank you for your explanation. Also, here's to Ubuntu!
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  9. #9
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    Hello, antz215!

    Another approach . . .


    If x=i is one root of x^3 + x^2 + x + 1 \:=\:0, find the other two roots.
    Multiply by (x-1)\!:\;\;(x-1)(x^3+x^2+x+1) \:=\:0\quad\Rightarrow\quad x^4-1\:=\:0

    . . x^4 \:=\:1\quad\Rightarrow\quad x^2 \:=\:\pm1 \quad\Rightarrow\quad x \:=\:\pm\sqrt{\pm1} \quad\Rightarrow\quad x \;=\;1,\,\text{-}1,\,i,\,\text{-}i


    We introduced x = 1; we were given x = i

    Therefore, the other two roots are: . x \;=\;\text{-}1,\,\text{-}i

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