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