will data indicate function is a 9th degree polynomial ?

Question

x1	x2=(x1)+1	x3= (x1)+2	(x4= (x1)+3	x5= (x1)+4	x6= (x1)+5	x7= (x1)+6	x8= (x1)+7	x9= (x1)+8	x10= (x1)+9	x11= (x1)+10
y1	y2	y3	y4	y5	y6	y7	y8	y9	y10	y11

given table of eleven x and y values where x is incremented by 1 for each entry after x1,

what linear relationship of y1,y2,...,y11 must be equal to zero for the data to be indicating y(x) is a polynomial of 9th degree, that is, indicates y(x)=ax⁹+bx⁸+cx⁷+dx⁶+ex⁵+fx⁴+gx³+hx²+jx+k?

given the data indicates 9th degree polynomial, what linear relationship of y1,y2,...,y10 divided by 9! equals the leading coefficient a?

Answer 1

To determine if a table of values represents a quadratic, we look at the 2nd difference. If the 2nd difference is a constant, then the relationship is a quadratic (2nd degree polynomial). So, consequently, if the 3rd difference is zero, then the relationship is a 2nd degree polynomial. Thusly, if the 10th difference is zero, then the relationship is a 9th degree polynomial.

If you make a generic "difference equation" using Y1, Y2, Y3, etc, the first difference would be (Y2 - Y1) or (Y3 - Y2) or (Y4 - Y3) etc. The 1st term of the second difference would be "(Y3 - Y2) - (Y2 - Y1)" or "Y3 - 2Y2 + Y1". The 1st term of the 3rd difference would be (Y4 - 3Y3 + 3Y2 - Y1) so we begin to see coefficients that mimic Pascals Triangle coefficients.

Pascals triangle has the following coefficients for the 10th row:

1 10 45 120 210 252 210 120 45 10 1

So we could extrapolate to say the 1st term of the 10th difference is:

Y11 - 10Y10 + 45Y9 - 120Y8 + 210Y7 - 252Y6 + 210Y5 - 120Y4 + 45Y3 - 10Y2 + Y1 and that difference would equal zero.

There's still some work to go on this, but it gives you a start.

Answer 2

The answer provide by William is correct and leads to two formulas.

Given data table described

then for n=1,3,5,7,9

data indicates nth degree polynomial, the first term being axⁿ if:

∑_i=0ⁿ⁺¹(-1)ⁱ*C_iⁿ⁺¹*y_i+1=0 which indicates nth degree polynomial, and

can also conclude

a=[∑_i=0ⁿ(-1)ⁱ⁺¹*C_iⁿ*y_i+1]/n!

then for n=2,4,6,8

data indicates nth degree polynomial, the first term being axⁿ if:

∑_i=0ⁿ⁺¹(-1)ⁱ⁺¹*C_iⁿ⁺¹*y_i+1=0 which indicates nth degree polynomial, and

can also conclude

a=[∑_i=0ⁿ(-1)ⁱ*C_iⁿ*y_i+1]/n!

here C_kⁿ=n!/[(n-k)!*k!]