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Math Help - The remainder and factor theorems

  1. #1
    Member Mr Rayon's Avatar
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    Arrow The remainder and factor theorems

    The remainder when x^2 - 3x + 1 is divided by (x + d) is 11. Find the possible values of d.
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    Quote Originally Posted by Mr Rayon View Post
    The remainder when x^2 - 3x + 1 is divided by (x + d) is 11. Find the possible values of d.
    Since the divisor is x + d, then the zero (when doing the synthetic division) is -d. So do the synthetic division:

    Code:
    -d | 1    -3         1
       |      -d    d^2+3d
       +------------------
         1  -d-3  d^2+3d+1
    Set the remainder equal to the given value:

    . . . . . d^2\, +\, 3d\, +\, 1\, =\, 11

    This rearranges as:

    . . . . . d^2\, +\, 3d\, -\, 10\, =\, 0

    Either factor the quadratic and then solve the factors for the values of d, or else plug the above into the Quadratic Formula.

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  3. #3
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    Hello, Mr Rayon!

    The remainder when x^2 - 3x + 1 is divided by (x + d) is 11.
    Find the possible values of d.
    From the heading, I assume you're expected to know the Remainder Theorem.

    . . If a polynomial p(x) is divided by (x-a), the remainder is p(a).


    We have: . p(x) \:=\:x^2-3x+1

    When p(x) is divided by (x+d), the remainder is p(\text{-}d)


    We have: . p(x) \:=\:x^2-3x+1 divided by (x-[\text{-}d])

    Hence: . (\text{-}d)^2 - 3(\text{-}d) + 1 \:=\:1 \quad\Rightarrow\quad d^2 + 3d \:=\:0 \quad\Rightarrow\quad d(d+3) \:=\:0

    Therefore: . d \;=\;0,\:\text{-}3

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  4. #4
    Member Mr Rayon's Avatar
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    Quote Originally Posted by Soroban View Post
    Hence: . (\text{-}d)^2 - 3(\text{-}d) + 1 \:=\:1
    ...But what happened to the 11?
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    Quote Originally Posted by Mr Rayon View Post
    ...But what happened to the 11?
    A small mistake by Soroban is what happened. But surely you can make the appropriate corrections .... Not that you need to do this, given Stapel's earlier post. Did you read it by the way?
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    Member Mr Rayon's Avatar
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    Quote Originally Posted by mr fantastic View Post
    A small mistake by Soroban is what happened. But surely you can make the appropriate corrections .... Not that you need to do this, given Stapel's earlier post. Did you read it by the way?
    Yes...I never skip a post




    d^2 + 3d - 10 = 0

    Using the quadratic formula we get:

    (-3 - 7)/2 = -5 or (-3 +7)/2 = 2

    But are there any other ways to solve this apart from by factorising?
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    Quote Originally Posted by Mr Rayon View Post
    But are there any other ways to solve this apart from by factorising?
    Yes; you can solve the quadratic by using the Quadratic Formula, or you can complete the square.
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  8. #8
    Member Mr Rayon's Avatar
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    Quote Originally Posted by stapel View Post
    Yes; you can solve the quadratic by using the Quadratic Formula, or you can complete the square.
    Yes...I tried to use completing the square but it got a bit messy down the end. I prefer using the quadratic formula.
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