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Algebra

# Polynomial Interpolation: Level 4 Challenges

$$f(x)$$ 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 $$\dfrac{a}{b}$$ 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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