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Math Help - more polynomials

  1. #1
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    Angry more polynomials

    Hi can someone please help me figure this out?

    WHen a polynomial is divided by (2x + 1)(x - 3), the remainder is 3x - 1. What is the remainder when the polynomial is divided by 2x +1?

    Thanks !
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  2. #2
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    Hello ipokeyou
    Quote Originally Posted by ipokeyou View Post
    Hi can someone please help me figure this out?

    WHen a polynomial is divided by (2x + 1)(x - 3), the remainder is 3x - 1. What is the remainder when the polynomial is divided by 2x +1?

    Thanks !
    The remainder when any polynomial in x is divided by a quadratic in x will be a linear expression in x: ax+b (where either or both of a and b may be zero); in other words the remainder will be of degree at most 1.

    When it is divided by a linear expression, the remainder will be a constant term (which, again, may be zero).

    So let's suppose that the polynomial is f(x) and that the quotient when it is divided by (2x+1)(x-3) is q(x). Then, since we know that the remainder is 3x-1, we can write:
    f(x) = (2x+1)(x-3)q(x) + 3x -1
    which we can write as:
    f(x) = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac32-1
    = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac52
    Now when this expression is divided by (2x+1) the quotient will be the polynomial (x-3)q(x) + \frac32, and the remainder will be -\frac52. So there's your answer: -\frac52=-2\tfrac12.

    Grandad
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