Algebra
# Polynomial Interpolation

$f(x)$ is a $5^\text{th}$-degree polynomial such that $f(1)=2,$ $f(2)=3,$ $f(3)=4,$ $f(4)=5,$ $f(5)=6,$ and $f(8)=7.$

If the value of $f(9)$ can be expressed as $\frac{a}{b}$ for coprime positive integers $a$ and $b$, find the value of $a+b$.

$f(x)$ is a polynomial with integer coefficients. We have,

$f(1)=1$

$f(2)=4$

$f(3)=9$

$f(4)=16$

$f(5)=25$

$f(6)=36$

$f(7)=49$

$f(8)=64$

$f(9)=81$

$f(10)=100$

Determine $f(11)$.

Given that $P(x)$ is a monic fifth-degree polynomial such that

$\begin{aligned} P(1) & = & 1^2 \\ P(2) & = & 2^2 \\ P(3) & = & 3^2 \\ P(4) & = & 4^2 \\ P(5) & = & 5^2,\end{aligned}$

find the value of $P(6)$.

Roopesh went to his friends house. There he ate a lot of Ice-cream. First he started with Vanilla then Chocolate then Vanilla then Butterscotch then Vanilla. He is a mathematician and made a monic polynomial of degree 5 which gave him values of the first letter of the Ice Cream.

This is the way his polynomial proceeded :-

$p(1) = 22$

$p(2) = 3$

$p(3) = 22$

$p(4) = 2$

$p(5) = 22$

If the value of $p(6) = \overline{abc}$. The first letter of the ice cream he eats would be $a+b+c$. Which ice cream will be eat next?

**Assumption:** $\overline{abc}$ represents a 3 digit number.

$\large f(1) = 4, f(2) = 9, f(3) = 20, f(4) = 44, f(5) = 88$

Let $f(x)$ be a 5th degree monic polynomial such that it satisfies the equations above. Find the value of $f(6)$.