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Math Help - finding polynomials

  1. #1
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    finding polynomials

    For the following, find polynomials q(x) and r(x) such that b(x) = q(x)a(x)+r(x), where r(x) = 0 or deg r(x) < deg a(x).

    (a) a(x) = x^2-2x+4, b(x) = 2x^5-x^4+3x^3-2x+1

    any help would be much appreciated.
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  2. #2
    Member arpitagarwal82's Avatar
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    Quote Originally Posted by jvignacio View Post
    For the following, find polynomials q(x) and r(x) such that b(x) = q(x)a(x)+r(x), where r(x) = 0 or deg r(x) < deg a(x).

    (a) a(x) = x^2-2x+4, b(x) = 2x^5-x^4+3x^3-2x+1

    any help would be much appreciated.
    deg r(x) < deg a(x).
    so r(x) must be of degree 1.

    b(x) is of degree 5, so q(x)a(x) must be of degree 5.
    so we get q(x) of degree 3.

    let q(x) = ax^3 + bx^2 + cx + d
    and
    r(x) = ex + f.

    so we have

    2x^5 - x^4 + 3x^3 - 2x + 1 = (x^2 - 2x + 4)(ax^3 + bx^2 + cx + d) + ex +f

    Now equate the co-efficient of each power of x from LHS and RHS.
    You will get 6 equations (linear0 and 6 variables. solve for a,b,c,d,e and f to get both polynomials.
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  3. #3
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    Quote Originally Posted by jvignacio View Post
    For the following, find polynomials q(x) and r(x) such that b(x) = q(x)a(x)+r(x), where r(x) = 0 or deg r(x) < deg a(x).

    (a) a(x) = x^2-2x+4, b(x) = 2x^5-x^4+3x^3-2x+1
    This is just asking you to use do the polynomial long division, and then solve for the needed information. This is because [bthe Remainder Theorem[/b] relates the division to the factoring, etc, etc, as:

    . . . . ."b(x) = q(x)a(x) + r(x)"
    . . . . .is the same as
    . . . . ."b(x)/a(x) = q(x) + r(x)/a(x)"

    (This is just like "13 = 2*5 + 3" meaning the same as "13/5 = 2 + 3/5", by the way.)

    [HTML] 2x^3 + 3x^2 + 1x - 10
    --------------------------------------
    x^2 - 2x + 4 )2x^5 - 1x^4 + 3x^3 + 0x^2 - 2x + 1
    2x^5 - 4x^4 + 8x^3
    --------------------------------------
    3x^4 - 5x^3 + 0x^2 - 2x + 1
    3x^4 - 6x^3 + 12x^2
    --------------------------------------
    1x^3 - 12x^2 - 2x + 1
    1x^3 - 2x^2 + 4x
    --------------------------------------
    -10x^2 - 6x + 1
    -10x^2 + 20x - 40
    --------------------------------------[/HTML]
    Since the remainder will obviously be linear (once you complete the division), r(x) will fit the requirements.
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