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Math Help - Remainder & Factor Thm

  1. #1
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    Remainder & Factor Thm

    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 remainder constant k when divided by (x-a)(x-b), k \neq 0.

    My proof:

    g(x)=(x-a)Q_{1}(x)+k
    g(x)=(x-b)Q_{2}(x)+k

    where Q_{1}(x) and Q_{2}(x) are quotients.

    Let f(x)=g(x)-k,
    then f(x)=(x-a)Q_{1}(x) and f(x)=(x-b)Q_{2}(x).

    So (x-a) and (x-b) are factors of f(x)

    Thus,
    f(x)=(x-a)(x-b)Q_{3}(x)
    \Rightarrow g(x)-k=(x-a)(x-b)Q_{3}(x)
    \Rightarrow g(x)=(x-a)(x-b)Q_{3}(x)+k

    Hence, it is TRUE.

    Is my proof ok? Is there a better one?
    Many thanks for those who help
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  2. #2
    Moo
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    Hello,

    Your proof is ok
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