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.

Thanks!!

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- Jul 19th 2007, 12:27 AMacc100jtQuestion 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.

Thanks!! - Jul 19th 2007, 01:00 AMred_dog
We have $\displaystyle g(a)=g(b)=k$

$\displaystyle g(x)=(x-a)(x-b)q(x)+\alpha x+\beta$

$\displaystyle g(a)=k\Rightarrow \alpha a+\beta =k$

$\displaystyle g(b)=k\Rightarrow \alpha b+\beta =k$

Then $\displaystyle \alpha (a-b)=0\Rightarrow \alpha =0\Rightarrow \beta =k$