Question 1119948
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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)\ =\ 0] when *[tex \Large x\ =\ 1], then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ +\ e\ +\ f\ =\ 0]


And if *[tex \Large f(x)\ =\ 3] when *[tex \Large x\ =\ 2], then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 32a\ +\ 16b\ +\ 8c\ +\ 4d\ +\ 2e\ +\ f\ =\ 3]


Similarly:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 243a\ +\ 81b\ +\ 27c\ +\ 9d\ +\ 3e\ +\ f\ =\ 16]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 1024a\ +\ 256b\ +\ 64c\ +\ 16d\ +\ 4e\ +\ f\ =\ 45]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3125a\ +\ 625b\ +\ 125c\ +\ 25d\ +\ 5e\ +\ f\ =\ 96]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 7776a\ +\ 1296b\ +\ 216c\ +\ 36d\ +\ 6e\ +\ f\ =\ 175]


Use Cramer's Rule to solve the 6X6 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 Da determinant will be:
<pre>
     |    1    1    1    1    1   1|
     |   32   16    8    4    2   1|
D  = |  243   81   27    9    3   1|
     | 1024  256   64   16    4   1|
     | 3125  625  125   25    5   1|
     | 7776 1296  216   36    6   1|

     |    0    1    1    1    1   1|
     |    3   16    8    4    2   1|
Da = |   16   81   27    9    3   1|
     |   45  256   64   16    4   1|
     |   96  625  125   25    5   1|
     |  175 1296  216   36    6   1|
</pre>


and so on...
								
								
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 \ \ \ \ \ \ \ \ \ \  
								
{{n}\choose{r}}

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