If a polynomial is divided by $(x-1)$ then remainder is 5 and if divided by $(x-2)$ the remainder is 7. What will be the remainder is the polynomial is divided by $(x-1)(x-2)$ ?
As the degree is unknown so we can't write the polynomial with arbitrary coefficients. So we have to assume the polynomial as $f(x)$. Now we can write ... $$f(x)=(x-1)g(x)+5$$ $$f(x)=(x-2)h(x)+7$$ Where $g(x)$ & $h(x)$ are some polynomial of x. then I subtract these two equations, but can't go further. Am I going correct? Should I need to use differentiation?
(If you are using some theorem please provide a link so that I can learn that)