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Thread: 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 $\displaystyle x$ is divided by a quadratic in $\displaystyle x$ will be a linear expression in $\displaystyle x$: $\displaystyle ax+b$ (where either or both of $\displaystyle a$ and $\displaystyle b$ 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 $\displaystyle f(x)$ and that the quotient when it is divided by $\displaystyle (2x+1)(x-3)$ is $\displaystyle q(x)$. Then, since we know that the remainder is $\displaystyle 3x-1$, we can write:
    $\displaystyle f(x) = (2x+1)(x-3)q(x) + 3x -1$
    which we can write as:
    $\displaystyle f(x) = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac32-1$
    $\displaystyle = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac52$
    Now when this expression is divided by $\displaystyle (2x+1)$ the quotient will be the polynomial $\displaystyle (x-3)q(x) + \frac32$, and the remainder will be $\displaystyle -\frac52$. So there's your answer: $\displaystyle -\frac52=-2\tfrac12$.

    Grandad
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