document.write( "Question 94361: I need help with two different charts:\r
\n" ); document.write( "\n" ); document.write( "n 1 2 3 4 5
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\n" ); document.write( "\n" ); document.write( "n 1 2 3 4 5
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\n" ); document.write( "\n" ); document.write( "Need to find degree and equation for the polynomial
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Algebra.Com's Answer #68724 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
Soluton by Edwin McCravy:
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\n" ); document.write( "Tutor Stanbon assumed you were taking statistics and were
\n" ); document.write( "looking for a regression equation. But you were not looking
\n" ); document.write( "for that. You want to find a polynomial f(n) that will have
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document.write( "I need help with two different charts:\r\n" );
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document.write( "n    1    2     3     4     5  \r\n" );
document.write( "f(n) 0    4     20    54    112\r\n" );
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document.write( "To find the degree:\r\n" );
document.write( "Make a difference table, by finding the\r\n" );
document.write( "difference of each adjacent pair, writing\r\n" );
document.write( "the differences until you find a line of\r\n" );
document.write( "differences that are all the same.\r\n" );
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document.write( "n    1     2      3     4     5  \r\n" );
document.write( "f(n) 0     4     20    54    112\r\n" );
document.write( "1       4     16    34    58\r\n" );
document.write( "2          12    18    24 \r\n" );
document.write( "3              6    6\r\n" );
document.write( "                 \r\n" );
document.write( "It took 3 lines of differences to find\r\n" );
document.write( "a line on which all the numbers are the \r\n" );
document.write( "same, so we know to assume a polynomial \r\n" );
document.write( "of degree 3:\r\n" );
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document.write( "f(n) = An³ + Bn² + Cn + D \r\n" );
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document.write( "Since f(1) = 0,\r\n" );
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document.write( "f(1) = A(1)³ + B(1)² + C(1) + D = 0\r\n" );
document.write( "f(1) = A + B + C + D = 0\r\n" );
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document.write( "So\r\n" );
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document.write( "Since f(2) = 4\r\n" );
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document.write( "f(2) = A(2)³ + B(2)² + C(2) + D = 4\r\n" );
document.write( "f(2) = A(8) + B(4) + C(2) + D = 4\r\n" );
document.write( "f(2) = 8A + 4B + 2C + D = 4\r\n" );
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document.write( "Since f(3) = 20\r\n" );
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document.write( "f(3) = A(3)³ + B(3)² + C(3) + D = 20\r\n" );
document.write( "f(3) = A(27) + B(9) + C(3) + D = 2-\r\n" );
document.write( "f(3) = 27A + 9B + 3C + D = 20\r\n" );
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document.write( "Since f(4) = 54\r\n" );
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document.write( "f(4) = A(4)³ + B(4)² + C(4) + D = 54\r\n" );
document.write( "f(3) = A(64) + B(16) + C(4) + D = 54\r\n" );
document.write( "f(3) = 64A + 16B + 4C + D = 54\r\n" );
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document.write( "So we have the system of equations:\r\n" );
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document.write( "  A +   B +  C + D =  0\r\n" );
document.write( " 8A +  4B + 2C + D =  4\r\n" );
document.write( "27A +  9B + 3C + D = 20\r\n" );
document.write( "64A + 16B + 4C + D = 54\r\n" );
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document.write( "Do you know how to solve that system of\r\n" );
document.write( "equations with matrices?  If not post\r\n" );
document.write( "again asking how.  The solution is\r\n" );
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document.write( "A=1, B=0, C=-3, D=2\r\n" );
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document.write( "So\r\n" );
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document.write( "f(n) = An³ + Bn² + Cn + D so\r\n" );
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document.write( "f(n) = 1n³ + 0n² + (-3)n + 2\r\n" );
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document.write( "f(n) = n³ - 3n + 2\r\n" );
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document.write( "------------------------------- \r\n" );
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document.write( "n    1     2     3     4     5\r\n" );
document.write( "f(n) 2    17    86    275   674\r\n" );
document.write( "\r\n" );
document.write( "Need to find degree and equation for the polynomial\r\n" );
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document.write( "Make a difference table, by finding the\r\n" );
document.write( "difference of each adjacent pair, writing\r\n" );
document.write( "the differences until you find a line of\r\n" );
document.write( "differences that are all the same.\r\n" );
document.write( "\r\n" );
document.write( "n    1     2      3     4     5  \r\n" );
document.write( "f(n) 2     17     86    275   674\r\n" );
document.write( "1       15     69   189    399\r\n" );
document.write( "2          54    120    210 \r\n" );
document.write( "3            66     90\r\n" );
document.write( "4                 24\r\n" );
document.write( "                  \r\n" );
document.write( "We had to make 4 lines this time because\r\n" );
document.write( "we didn't come to a row that was all the\r\n" );
document.write( "same, so we had to take it all the way to\r\n" );
document.write( "4 lines, so the degree is 4.\r\n" );
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document.write( "f(n) = An4 + Bn³ + Cn² + Dn + E\r\n" );
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document.write( "Since f(1) = 2,\r\n" );
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document.write( "f(1) = A(1)4 + B(1)³ + C(1)² + D(1) + E = 2\r\n" );
document.write( "f(1) = A + B + C + D + E = 2\r\n" );
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document.write( "Since f(2) = 17\r\n" );
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document.write( "f(2) = A(2)4 + B(2)³ + C(2)² + D(2) + E = 17\r\n" );
document.write( "f(2) = A(16) + B(8) + C(4) + D(2) + E = 17\r\n" );
document.write( "f(2) = 16A + 8B + 4C + 2D + E = 17\r\n" );
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document.write( "Since f(3) = 86\r\n" );
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document.write( "f(3) = A(3)4 + B(3)³ + C(3)² + D(3) + E = 86\r\n" );
document.write( "f(3) = A(81) + B(27) + C(9) + 3D + E = 86\r\n" );
document.write( "f(3) = 81A + 27B + 9C + 3D + E = 86\r\n" );
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document.write( "Since f(4) = 275\r\n" );
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document.write( "f(4) = A(4)4 + B(4)³ + C(4)² + D(4) + E = 275\r\n" );
document.write( "f(4) = A(256) + B(64) + C(16) + 4D + E = 275\r\n" );
document.write( "f(4) = 256A + 64B + 16C + 4D + E = 275\r\n" );
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document.write( "Since f(5) = 674\r\n" );
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document.write( "f(4) = A(5)4 + B(5)³ + C(5)² + D(5) + E = 674\r\n" );
document.write( "f(4) = A(625) + B(125) + C(25) + 5D + E = 674\r\n" );
document.write( "f(4) = 625A + 125B + 25C + 5D + E = 674\r\n" );
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document.write( "So we have the system of equations:\r\n" );
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document.write( "   A +    B +   C +  D + E =   2\r\n" );
document.write( " 16A +   8B +  4C + 2D + E =  17 \r\n" );
document.write( " 81A +  27B +  9C + 3D + E =  86\r\n" );
document.write( "256A +  64B + 16C + 4D + E = 275\r\n" );
document.write( "625A + 125B + 25C + 5D + E = 674\r\n" );
document.write( "\r\n" );
document.write( "Do you know how to solve that system of\r\n" );
document.write( "equations with matrices?  If not post\r\n" );
document.write( "again asking how.  The solution is\r\n" );
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document.write( "A=1, B=1, C=-4, D=5, E=-1\r\n" );
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document.write( "So\r\n" );
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document.write( "f(n) = An4 + Bn³ + Cn² + Dn + E\r\n" );
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document.write( "f(n) = 1n4 + 1n³ + (-4)n² + 5n - 1\r\n" );
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document.write( "f(n) = n4 + n³ - 4n² + 5n - 1\r\n" );
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document.write( "Edwin
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