Question

Find the lowest order a polynomial could have that passes through the points (1, 20), (2, 49), (3, 102), (4, 185), (5, 304). Don't just give me the answer please! First of all, what does it mean by order?

Answer 1

Order of a polynomial is the highest power of x in the polynomial.
Like in
\[x^3+3x^2+2x+2\]
it's 3 here

Answer 2

So order is another name for degree?

Answer 3

yeah

Answer 4

I'm not really sure what to do. It seems like an exponential function?

Answer 5

Just a moment, I'm trying

Answer 6

Are you studying LaGrange polynomials? That's one way to do it (a lot to write though).

Answer 7

Ummm no? This is apparently advanced Algebra II.

Answer 8

Oh wait. I see what I have to do. It says lowest order right? Since it says it's a polynomial, the only kind of polynomial order that can create this is one that is odd numbered right? Then it would be 3 because it is the lowest odd number that isn't 1 because a degree of one is a linear line. Does that make sense?

Answer 9

I just need someone to verify if it makes sense for me.

Answer 10

I can use LaGrange to figure out the actual equation, but that may not help you on a test (especially because LaGrange just takes a long time to write out). Is this a question from part of a chapter? If so, what is the chapter about?

Answer 11

This is from a Math Team Test. Topic Applications meaning word problems.

Answer 12

I'm just wondering if my logic above makes sense because they're not asking for anything more than the lowest possible degree...

Answer 13

Ok, I know the equation, but I have no idea how it ties into your class. Where are you getting "the only kind of polynomial order that can create this is one that is odd numbered right? Then it would be 3"

Answer 14

The end results of polynomials are different if the degree is even or odd right? An odd degree gives you a graph going infinitely in opposite directions?

Answer 15

THis is the explanation they gave. Maybe it'll help? Using the method of finite differences, the first-order differences are 29, 53, 83, and 119; the second-order differences are 24, 30, and 36; and the third-order differences are 6 and 6. Therefore, the polynomial could possibly have degree 3.

Answer 16

Let me refresh myself in finite differences real quick (but yes, the formula is a 3rd degree polynomial).

Answer 17

Found a site online that calculates LaGrange (schweet!). It's saying the polynomial would be:
\[x^{3} + 6x^{2} + 4x + 9\]

Answer 18

Yeah, for an nth order polynomial nth order difference is constant, here 3rd order difference is constant.
Indeed it's a 3rd order

Answer 19

Can you explain what the method of finite differences is or at least link me somewhere that explains it well?

Answer 20

Please see this, they have explained it properly
http://www.jimloy.com/algebra/finite.htm

Answer 21

Yes, Lagrange interpolation is one way to solve this problem : http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

Answer 22

There are different other methods, but I am not sure if this is solvable outside the domain of numerical analysis.

Answer 23

@FoolForMath Found this site that calculates LaGrange for you: http://wood.mendelu.cz/math/maw-html/index.php?lang=en&form=lagrange

Answer 24

@ash2326 Thanks for the link, cleared the cobwebs :) Haven't seen that in waaaay too long lol!

Answer 25

Glad to help @Wired

Answer 26

Thanks everyone!

Answer 27

For those who read this later:


(1, 20), (2, 49), (3, 102), (4, 185), (5, 304)
x y diff1 diff2 diff3
1 20
2 49 29
3 102 53 24
4 185 83 30 6
5 304 119 36 6