[ Lagrange Interpolation Polynomial ]

Construct polynomials $f(x)$ of degree not higher than 2 that satisfy the conditions:
a) $f(0)=1, f(1)=3, f(2)=3$

b) $f(-1)=-1, f(0)=2, f(1)=5$

c) $f(-1)=1, f(0)=0, f(2)=4$.

#
a) By the formula from problem $\underline{61052}$ (or $\underline{61053}$)

$$
f(x)=\frac{(x-1)(x-2)}{1 \cdot 2}-3 \frac{x(x-2)}{1 \cdot 1}+3 \frac{x(x-1)}{2 \cdot 1}=-x^{2}+3 x+1
$$

b) $f(x)=-\frac{x(x-1)}{1 \cdot 2}-2 \frac{(x+1)(x-1)}{1 \cdot 1}+5 \frac{(x+1) x}{2 \cdot 1}=3 x+2$.

c) $f(x)=\frac{x(x-2)}{1 \cdot 3}+4 \frac{(x+1) x}{3 \cdot 2}=x^{2}$.

## Answer

a) $-x^{2}+3 x+1$
b) $3 x+2$;
c) $x^{2}$.