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

\[ \begin{eqnarray} P(1) & = & 1^2 \\ P(2) & = & 2^2 \\ P(3) & = & 3^2 \\ P(4) & = & 4^2 \\ P(5) & = & 5^2,\end{eqnarray} \]

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 \(f(6)\).

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