Interpolation Polynomial of Lagrange, Theorem of Bezout. Factorization

Let $A, B$ and $C$ be the remainders when the polynomial $P(x)$ is divided by $x-a, x-b$, and $x-c$.

Find the remainder when the same polynomial is divided by the product $(x-a)(x-b)(x-c)$.
Let $R(x)$ be the desired remainder. Then this is a polynomial of degree no higher than two, which, according to the theorem of Bezout, at points $a, b$, and $c$ takes the values $A, B$, and $C$ respectively. Therefore, $R(x)$ is uniquely determined by the formula from problem $\underline{61052 .}$

## Answer

$A \frac{(x-b)(x-c)}{(a-b)(a-c)}+B \frac{(x-a)(x-c)}{(b-a)(b-c)}+C \frac{(x-a)(x-b)}{(c-a)(c-b)}$.