Question 1120309
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Six points uniquely determine a 5th-degree polynomial function.  (In general, n points uniquely determine an (n-1)th degree polynomial function).


The general form of a 5th-degree polynomial function is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ ax^5\ +\ bx^4\ +\ cx^3\ +\ dx^2\ +\ ex\ +\ f]


So if *[tex \Large f(x)\ =\ 3] when *[tex \Large x\ =\ 0], then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\ =\ 3]


We can use this result to reduce the linear system from a 6X6 system to a 5X5 system, as follows


If *[tex \Large f(x)\ =\ 9] when *[tex \Large x\ =\ 1], then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  a\ +\  b\ +\  c\ +\  d\ +\  e\ =\ 6]


Taking into account the fact that *[tex \Large\ f\ =\ 3]


Similarly:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 32a\ +\ 16b\ +\ 8c\ +\ 4d\ +\ 2e\ =\ 9]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 243a\ +\ 81b\ +\ 27c\ +\ 9d\ +\ 3e\ =\ 18]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 1024a\ +\ 256b\ +\ 64c\ +\ 16d\ +\ 4e\ =\ 30]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3125a\ +\ 625b\ +\ 125c\ +\ 25d\ +\ 5e\ =\ 51]


Use Cramer's Rule to solve the 5X5 system.  Easiest is to put your matrices into Excel and use the MDETERM function to calculate the required determinants.


For example, your coefficient determinant and D_a determinant will be:
<pre>
     |    1    1    1    1    1 |
     |   32   16    8    4    2 |
D  = |  243   81   27    9    3 |
     | 1024  256   64   16    4 |
     | 3125  625  125   25    5 |

     |    6    1    1    1    1 |
     |    9   16    8    4    2 |
D_a= |   18   81   27    9    3 |
     |   30  256   64   16    4 |
     |   51  625  125   25    5 |
</pre>


and so on...


You can also find an online matrix calculator that will do Gauss-Jordan reduction for you.  The only alternative to getting the exact answer by solving the 5X5 linear system is to use a "curve of best fit" calculator that will handle a 5th-degree polynomial model.  These generally result in inexact but very precise coefficients.


Once you have the coefficients of the 5th-degree polynomial function that models your data, you can evaluate *[tex \Large f(6)\ \ ] and *[tex \Large\ f(7)] to find the next two numbers in the sequence.
								
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
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