Polynomials & Rational Functions
Review question
What if dividing $f(x)$ by $(x-a)(x-b)$ and by $(x-a)(x-c)$ gives the same remainder?
Ref: R8682
Question
Show that the remainder when the polynomial \(f(x)\) is divided by \((x-a)\) is \(f(a)\). Show further that, if \(f(x)\) is divided by \((x-a)(x-b)\), where \(a \ne b\), then the remainder is
\[\begin{equation*}
\left( \frac{f(a) - f(b)}{a-b} \right)x + \left( \frac{af(b) - bf(a)}{a-b} \right).
\end{equation*}\]
The remainders when \(f(x)\) is divided by \((x-a)(x-b)\) and by \((x-a)(x-c)\), \(a \ne b \ne c\), are equal. Prove that
\[\begin{equation*}
(b-c)f(a) + (c-a)f(b) + (a-b)f(c) = 0.
\end{equation*}\]
