# Question on Remainder Theorem

• Jul 19th 2007, 12:27 AM
acc100jt
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.

Thanks!!
• Jul 19th 2007, 01:00 AM
red_dog
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 $\alpha (a-b)=0\Rightarrow \alpha =0\Rightarrow \beta =k$