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Math Help - Question on Remainder Theorem

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    Question on Remainder Theorem

    The expression g(x) has the same remainder constant k when divided by
    (x-a) or (x-b). Determine, with reasons, whether it is always true that g(x) also has reminder constant k when divided by (x-a)(x-b), k not equal to 0.

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  2. #2
    MHF Contributor red_dog's Avatar
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    We have g(a)=g(b)=k

    g(x)=(x-a)(x-b)q(x)+\alpha x+\beta
    g(a)=k\Rightarrow \alpha a+\beta =k
    g(b)=k\Rightarrow \alpha b+\beta =k
    Then \alpha (a-b)=0\Rightarrow \alpha =0\Rightarrow \beta =k
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