Algebra

Polynomial Interpolation

Polynomial Interpolation: Level 4 Challenges

         

f(x)f(x) is a 5th5^\text{th}-degree polynomial such that f(1)=2,f(1)=2, f(2)=3,f(2)=3, f(3)=4,f(3)=4, f(4)=5,f(4)=5, f(5)=6,f(5)=6, and f(8)=7.f(8)=7.

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

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

f(1)=1f(1)=1

f(2)=4f(2)=4

f(3)=9f(3)=9

f(4)=16f(4)=16

f(5)=25f(5)=25

f(6)=36f(6)=36

f(7)=49f(7)=49

f(8)=64f(8)=64

f(9)=81f(9)=81

f(10)=100f(10)=100

Determine f(11)f(11).

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

P(1)=12P(2)=22P(3)=32P(4)=42P(5)=52, \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)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)=22p(1) = 22
p(2)=3p(2) = 3
p(3)=22p(3) = 22
p(4)=2p(4) = 2
p(5)=22p(5) = 22

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

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

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

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

×

Problem Loading...

Note Loading...

Set Loading...