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Math Help - Cubic Eqations

  1. #1
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    Cubic Eqations

    1. Determine the remainder when 9x5 – 4x4 is divided by 3x-1


    2. Show, using factor theorem, that 2x-1 is a factor of:


    2x4 - x3-6x2 + 5x -1
    and hence express 2x4 - x3-6x2 + 5x -1 as a product of a linear and cubic factor


    Where do I start?
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  2. #2
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    1. Determine the remainder when 9x5 – 4x4 is divided by 3x-1
    You can use either the polynomial long division, or the polynomial remainder theorem. According to the latter, the remainder of f(x) divided by k(x-a) is f(a).

    2. Show, using factor theorem, that 2x-1 is a factor of:


    2x4 - x3-6x2 + 5x -1
    and hence express 2x4 - x3-6x2 + 5x -1 as a product of a linear and cubic factor
    According to the polynomial remainder theorem above (in this case it is the same as the factor theorem), it is sufficient to represent 2x-1 as k(x - a) for some k and a and then to show that a is the root of the given polynomial. To find the cubic factor, use the long division to divide the given fourth-degree polynomial by 2x - 1.
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  3. #3
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    Quote Originally Posted by DanBrown100 View Post
    1. Determine the remainder when 9x5 – 4x4 is divided by 3x-1


    2. Show, using factor theorem, that 2x-1 is a factor of:


    2x4 - x3-6x2 + 5x -1
    and hence express 2x4 - x3-6x2 + 5x -1 as a product of a linear and cubic factor


    Where do I start?
    1. \displaystyle f(x) = 9x^5 - 4x^4.

    The remainder when divided by \displaystyle 3x - 1 is given by \displaystyle f\left(\frac{1}{3}\right).


    2. \displaystyle f(x) = 2x^4 - x^3 - 6x^2 + 5x - 1.

    To show that \displaystyle 2x - 1 is a factor, check that \displaystyle f\left(\frac{1}{2}\right) = 0.

    Then you need to use long division to express \displaystyle f(x) as \displaystyle (2x-1)Q(x).
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