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Math Help - Help forming polynomial of least degree

  1. #1
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    Help forming polynomial of least degree

    5. b) Find a real polynomial of least degree having 1+i, 2, -i as roots.

    My work:

    1.
    x=1+i
    x=2
    x= -i

    2.
    [(x-1)-i]
    [(x-2)]
    [(x)+i]


    3.
    Secondly I tried putting them together and got:

    (x-1)(x) + (x-1)i - (x)i - i^2

    (x-1)(x) + (x-1)i - (x)i + 1

    (x-1)(x) - i + 1

    I cannot remove the i, please help.
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  2. #2
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    Quote Originally Posted by thekrown View Post
    5. b) Find a real polynomial of least degree having 1+i, 2, -i as roots.

    My work:

    1.
    x=1+i
    x=2
    x= -i

    2.
    [(x-1)-i]
    [(x-2)]
    [(x)+i]


    3.
    Secondly I tried putting them together and got:

    (x-1)(x) + (x-1)i - (x)i - i^2

    (x-1)(x) + (x-1)i - (x)i + 1

    (x-1)(x) - i + 1

    I cannot remove the i, please help.
    The roots of a polynomial whose coefficients are all real always occur in conjugate pairs ....
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  3. #3
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    Okay so we would have...

    [(x-1)-i]
    [(x-1)+i]

    [x-2]

    [(x)+i]
    [(x)-i]

    and then I would form the poly with these?
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  4. #4
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    Quote Originally Posted by thekrown View Post
    Okay so we would have...

    [(x-1)-i]
    [(x-1)+i]

    [x-2]

    [(x)+i]
    [(x)-i]

    and then I would form the poly with these?
    Take the product of these factors.

    Note that ([x - 1] - i)([x - 1] + i) = (x - 1)^2 + 1 = x^2 - 2x + 2 etc.
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  5. #5
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    I end up with [(x-1)^2+1] * [x-2] * [x^2+1]

    Is there a technique to shorten the multiplication process?
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  6. #6
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    Quote Originally Posted by thekrown View Post
    I end up with [(x-1)^2+1] * [x-2] * [x^2+1]

    Is there a technique to shorten the multiplication process?
    Why expand? Does the question ask you to expand? This is a correct answer. Unless you have good reason to do otherwise, leave it at that.
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  7. #7
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    The question is exactly as I typed above, so no they do not ask to expand. Good thing too.

    I always thought polynomials to be expanded as you say.

    Well thank you, you helped me a lot.
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