document.write( "Question 1120017: Could someone please walk me through how to do this. It is a question from my test prep of advanced functions. \r
\n" ); document.write( "\n" ); document.write( "How do I determine the equation of the function that would model this data?\r
\n" ); document.write( "\n" ); document.write( "x 1,2,3,4,5,6
\n" ); document.write( "f(x) 0,3,16,45,96,175\r
\n" ); document.write( "\n" ); document.write( "Any help would be so appreciated!!! \n" ); document.write( "
Algebra.Com's Answer #735692 by greenestamps(13198)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Use the method of finite differences to find the required degree of the polynomial. 6 points define a unique polynomial of degree 5; but it is possible a polynomial of lower degree will define the given points.
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document.write( "  0   3  16  45  96  175\r\n" );
document.write( "    3  13  29  51  79\r\n" );
document.write( "     10  16  22  28\r\n" );
document.write( "        6   6   6


\n" ); document.write( "The third row of differences is constant; that means the points can be defined by a polynomial of degree 3:

\n" ); document.write( "t(n) = an^3+bn^2+cn+d

\n" ); document.write( "Substitute n = 1, 2, 3, and 4 to get four equations in the coefficients a, b, c, and d and solve the system. Note it will always be easy to see what the next step should be in solving the system.
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document.write( "    a +   b +  c + d = 0\r\n" );
document.write( "   8a +  4b + 2c + d = 3\r\n" );
document.write( "  27a +  9b + 3c + d = 16\r\n" );
document.write( "  64a + 16b + 4c + d = 45
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document.write( "Comparing successive pairs of equations...:\r\n" );
document.write( "   7a +  3b +  c = 3\r\n" );
document.write( "  19a +  5b +  c = 13\r\n" );
document.write( "  37a +  7b +  c = 29
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document.write( "Again comparing successive pairs of equations...:\r\n" );
document.write( "  12a +  2b = 10\r\n" );
document.write( "  18a +  2b = 16
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document.write( "And comparing those two equations...:\r\n" );
document.write( "   6a = 6\r\n" );
document.write( "    a = 1
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document.write( "Then back substitute the known values to find the others:\r\n" );
document.write( "  12(1)+2b = 10\r\n" );
document.write( "  2b = -2\r\n" );
document.write( "  b = -1
\r\n" );
document.write( "  7(1)+3(-1)+c = 3\r\n" );
document.write( "  c = -1
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document.write( "  1+(-1)+(-1)+d = 0\r\n" );
document.write( "  d = 1

\n" ); document.write( "We have a=1, b=-1, c=-1 and d=1; the polynomial of degree 3 that defines the given numbers is

\n" ); document.write( "t(n) = n^3-n^2-n+1
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