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Math Help - Three tricky questions that need solving..

  1. #1
    Member MathBlaster47's Avatar
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    Three tricky questions that need solving..

    Hello MHF!
    I have three questions that which are throwing me for a loop. While I understand some of the principals behind them, the questions themselves defy my ability to work out.

    Q1:
    Simplify:
     \frac{1}{2-i}
    (I have no idea where to begin to work this one out, or even if I have to...)

    Q2:Find a quadratic equation with the roots (4+i) and (4-i).
    (I have no idea where to start with this one either.......)

    Q3:Use the factor theorem to determine whether (x-3) is a factor of f(x)=x^4+12x^3+6x+27
    (I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

    Many thanks in advance!
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  2. #2
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    pickslides's Avatar
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    Hi there MathBlaster47

    Quote Originally Posted by MathBlaster47 View Post

    Q1:
    Simplify:
     \frac{1}{2-i}
    (I have no idea where to begin to work this one out, or even if I have to...)
    Using the complex conjugate, expand the following

     \frac{1}{2-i}\times\frac{2+i}{2+i}

    Quote Originally Posted by MathBlaster47 View Post



    Q2:Find a quadratic equation with the roots (4+i) and (4-i).
    (I have no idea where to start with this one either.......)

    expand (x-(4+i))(x-(4-i))



    Quote Originally Posted by MathBlaster47 View Post

    Q3:Use the factor theorem to determine whether (x-3) is a factor of f(x)=x^4+12x^3+6x+27
    (I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

    will be a factor if f(3)=0

    now complete f(3)=3^4+12\times 3^3+6\times 3+27

    is this zero?
    Last edited by pickslides; January 5th 2010 at 01:28 PM. Reason: bad latex
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  3. #3
    Member MathBlaster47's Avatar
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    Quote Originally Posted by pickslides View Post
    Hi there MathBlaster47



    Using the complex conjugate, expand the following

     \frac{1}{2-i}\times\frac{2+i}{2+i}



    expand (x-(4+i))(x-(4-i))






    will be a factor if f(3)=0

    now complete f(3)=3^4+12\times 3^3+6\times 3+27

    is this zero?
    I Thank you heartily Pickslides!

    "expand (x-(4+i))(x-(4-i))"

    So does it become: x^2-8 x+17 ?
    Last edited by MathBlaster47; January 5th 2010 at 02:13 PM.
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  4. #4
    Master Of Puppets
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    Quote Originally Posted by MathBlaster47 View Post
    I Thank you heartily Pickslides!

    "expand (x-(4+i))(x-(4-i))"

    So does it become: x^2-8 x+17 ?
    I like it.
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  5. #5
    Member MathBlaster47's Avatar
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    Fantastic!
    Thank you once more!
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  6. #6
    Member MathBlaster47's Avatar
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    Quote Originally Posted by pickslides View Post

    Quote:
    Originally Posted by MathBlaster47

    Q3:Use the factor theorem to determine whether (x-3) is a factor of
    (I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)



    will be a factor if

    now complete

    is this zero?

    A quick followup question:
    If I want to do synthetic division on this function, would I be using 3 or -3?
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  7. #7
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by MathBlaster47 View Post
    A quick followup question:
    If I want to do synthetic division on this function, would I be using 3 or -3?
    You'd be dividing f(x) by (x-3). I never learnt synthetic division so I can't tell you how you'd work it out
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  8. #8
    Member MathBlaster47's Avatar
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    bad type setting on my part.

    Quote Originally Posted by e^(i*pi) View Post
    You'd be dividing f(x) by (x-3). I never learnt synthetic division so I can't tell you how you'd work it out
    Well....I remember that when doing synthetic division the non-variable aspect of the divisor is used to "divide" the coefficients, but I can't remember if I am to use 3 or -3 in the place of the 'r'.
    3|
    1 12 0 6 27
    -3 -27 81 -261
    _________________
    1 9 -27 87 |-234


    or, 3|
    1 12 0 6 27
    3 45 135 423
    __________________
    1 15 45 141 | 450

    I know that the second version's remainder coincides with the function as worked with x=3. Therefore I am guessing that the second version is correct from what I understand of synthetic division and the remainder theorem, but I just want to make sure.

    Because I worked the function as f(3) and it does not equal 0, I think that means that (x-3) is not a factor, right?
    Last edited by MathBlaster47; January 6th 2010 at 04:01 PM. Reason: restoring working to the post
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