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Math Help - Please help to find k and remainder

  1. #1
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    Please help to find k and remainder

    Problem 1: Find k such that f(x)=x^4-kx^3+kx^2+1 has the factor (x+2)
    Problem 2: What is the remainder when f(x)=-3x^17+x^9-X^5+2x is divided by (x+1)

    Thank you
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  2. #2
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    Hello, oceanmd!

    I believe you're expected to know the Factor Theorem and Remainder Theorem.


    Problem 1: Find k such that f(x)\:=\:x^4-kx^3+kx^2+1 has the factor (x+2)

    If (x+2) is a factor of f(x), then: . f(\text{-}2) \:=\:0

    Hence: . f(\text{-}2) \;=\;(\text{-}2)^4 - k(\text{-}2)^3 + k(\text{-}2)^2 + 1 \;=\;0

    . . and we have: . 16 + 8k + 4k + 1 \:=\:0\quad\Rightarrow\quad12k \:=\:-17\quad\Rightarrow\quad\boxed{k \:=\:-\frac{17}{12}}



    Problem 2: What is the remainder when f(x)\:=\:-3x^{17}+x^9-x^5+2x is divided by (x+1)
    When f(x) is divided by (x+1), the remainder is: . f(\text{-}1)

    . . f(\text{-}1) \;=\;-3(\text{-}1)^{17} + (\text{-}1)^9 - (\text{-}1)^5 + 2(\text{-}1) \;=\;3 - 1 + 1 - 2 \;=\;\boxed{1}

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  3. #3
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    Thank you very much

    Soroban,

    Thanks a lot. These problems are from the Factor Theorem and Remainder Theorem Chapter.

    Have a nice rest of the day.
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