Given three remainders, what is the function?

Given three remainders, what is the function?

Question

A certain function of $x$ can be expressed in the form $a+b(x-1)+c(x-1)(x-2)+2(x-1)(x-2)(x-3),$ where $a$, $b$, $c$, are numerical. When the function is divided in turn by $x-1$, $x-2$, $x-3$ the remainders are $10$, $3$, $4$. Calculate the values of $a$, $b$, $c$.

Solution

Let

$f(x)$

denote our function of

$x$

. That is,

$$\begin{equation} f(x) = a+b(x-1)+c(x-1)(x-2)+2(x-1)(x-2)(x-3). \end{equation}$$

As $f(x)$ is a polynomial, the Remainder Theorem tells us that the remainder when divided by $(x-a)$ is equal to $f(a)$. Substituting $1$, $2$ and $3$ into $(1)$ in turn, we get that $f(1) = a,\quad f(2)=a+b \quad \text{and} \quad f(3)=a+2b+2c,$ and so $a=10$, $a+b=3$ and $a+2b+2c=4$.

Solving these equations then gives that $a=10$, $b=-7$ and $c=4$.