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Math Help - Polynomial help!!

  1. #1
    Newbie sir nerdalot's Avatar
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    Polynomial help!!

    I've been given this question as homework and i dont really understand the steps/how to work it out!! it is...

    Two integers are said to be relatively prime if their highest common factor is 1. If 'a' and 'b' are relatively prime it is possible to find integer 'x' and 'y' such that ax + by = 1.

    For example... 51 and 44 are relatively prime.

    Repeated use of the division identity leads to:

    51 = 44 x 1 + 7
    44 = 7 x 6 + 2
    7 = 3 x 2 + 1

    Reversing these steps leads to:

    1 = 7 - 3 x 2
    = 7 - 3(44 - 7 x 6)
    = 19 x 7 - 3 x 44
    = 19(51 - 44 x 1) - 3 x 44
    = 19 x 51 - 22 x 44

    question...

    a) use this method to find integers 'a' and 'b' such that 87a + 19b = 1.

    b) using the method, find the polynomials, A(x) and B(x) such that:
    A(x)(x(squared) - x) + B(x)(x(power of 4) + 4x(squared) - 4x + 4) = 1

    it's mainly the part in bold that i dont really get but i also dont know how to find the answer!!

    thanks tons!!
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  2. #2
    MHF Contributor
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    Division with remainders?

    1) 87/19 = 4 R 11 so 87 = 19*4 + 11 We're done with 87, but 19 and 11 moves on.

    2) 19/11 = 1 R 8 so 19 = 11*1 + 8 We're done with 19, but 11 and 8 move on.

    3) 11/8 = 1 R 3 so 11 = 8*1 + 3 We're done with 11, but 8 and 3 move on.

    4) 8/3 = 2 R 2 so 8 = 3*2 + 2 We're done with 8, but 3 and 2 move on.

    5) 3/2 = 1 R 1 so 3 = 2*1 + 1 Whew!! We finally found a 1!

    Now backwards. I'll let you do it, but with one hint. RESIST the temptation to simplify as you go. You may want to do a couple of things, but remember that the point of the exercise is to UNsimplify the expression.

    It shouldn't be too different for the polynomials.
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  3. #3
    Newbie sir nerdalot's Avatar
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    thanks for that... but how would i do part b??? i cant really see how to use the same pattern...

    thanks again!!
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  4. #4
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    It's only slightly more complicated. Remember that we use generally a "Base 10 Positional System", 1423 = 1*10^{3} + 4*10^{2} + 2*10^{1} + 3. If you think of the polynomials as a "Base 'x' Positional System" you are on your way. The ONLY significant complication is that you can get negative coefficients. You can't do that with non-polynomial constructs. 1(-4)23 would look pretty weird, but it DOES work that way, sort of, in Roman Numerals. "VL" is 45 -- That "V" is a negative value when it appears out of descending order. Just move on with long divisions as let's see what you get.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sir nerdalot View Post

    For example... 51 and 44 are relatively prime.

    Repeated use of the division identity leads to:

    51 = 44 x 1 + 7
    44 = 7 x 6 + 2
    7 = 3 x 2 + 1

    Reversing these steps leads to:

    1 = 7 - 3 x 2
    = 7 - 3(44 - 7 x 6)
    = 19 x 7 - 3 x 44
    = 19(51 - 44 x 1) - 3 x 44
    = 19 x 51 - 22 x 44

    question...

    a) use this method to find integers 'a' and 'b' such that 87a + 19b = 1.

    b) using the method, find the polynomials, A(x) and B(x) such that:
    A(x)(x(squared) - x) + B(x)(x(power of 4) + 4x(squared) - 4x + 4) = 1
    Quote Originally Posted by sir nerdalot View Post
    thanks for that... but how would i do part b??? i cant really see how to use the same pattern...
    It's "simple" (but ugly to write out)

    You start by taking x^2 - x and x^4 + 4x^2 - 4x + 4

    So:
    x^4 + 4x^2 - 4x + 4 = (x^2 - x)(x^2 + x + 5) + (x + 4)

    x^2 - x = (x + 4)(x - 5) + 20

    Now, in the problem with the integers we wanted a problem that ended with a 1. For the polynomials we want a problem that ends with an 0th degree polynomial, such as 20. So we are done here.

    Now reverse the process:
    20 = (x^2 - x) - (x + 4)(x - 5)

    20 = (x^2 - x) - ([(x^4 + 4x^2 - 4x + 4) - <br />
(x^2 - x)(x^2 + x + 5)])(x - 5)

    Expand:
    20 = (x^2 - x) - (x^4 + 4x^2 - 4x + 4)(x - 5) + <br />
(x^2 - x)(x^2 + x + 5)(x - 5)

    Collect terms:
    20 = (x^2 - x)[1 + (x^2 + x + 5)(x - 5)] - (x^4 + 4x^2 - 4x + 4)(x - 5)

    20 = (x^2 - x)(x^3 - 4x^2 - 24) + (x^4 + 4x^2 - 4x + 4)(-x + 5)<-- We are supposed to be adding the last term, so I changed the sign on the last polynomial.

    We almost have what we wanted. We wanted the RHS to equal 1. But hey, all we need to do now is divide both sides by 20:
    1 = (x^2 - x)\left ( \frac{1}{20}x^3 - \frac{1}{5}x^2 - \frac{6}{5} \right ) + (x^4 + 4x^2 - 4x + 4) \left ( -\frac{1}{20}x + \frac{1}{4} \right )

    Voila! It's exactly the same problem as with the integers done by exactly the same method. (Compare them.) It's just that the "book-keeping" is a tad messier.

    -Dan
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