# Math Help - Remainder & Factor Thm

1. ## 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

2. Hello,