document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=791951>Question 791951</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the equation of the polynomial function that passes through these points: (1,12), (2,-10), (3,-18), (4,0), (5,56), (6,162). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #479816 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20062)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=479816\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "To find the degree of the polynomial required, we \r\n" );
document.write( "make a difference table.  To do that we begin by\r\n" );
document.write( "making a list of the y-values in order of successive\r\n" );
document.write( "values of x\r\n" );
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document.write( "x   y\r\n" );
document.write( "1  12\r\n" );
document.write( "2 -10\r\n" );
document.write( "3 -18\r\n" );
document.write( "4   0\r\n" );
document.write( "5  56\r\n" );
document.write( "6 162\r\n" );
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document.write( "Then out to the right of that we make a list by\r\n" );
document.write( "subtracting each number in the y-list from the\r\n" );
document.write( "number just under it, except the last one.  That\r\n" );
document.write( "is we subtract \r\n" );
document.write( "(-10)-(12) = -22, then \r\n" );
document.write( "(-18)-(-10) = -18+10 = -8, etc:\r\n" );
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document.write( "1  12  -22    \r\n" );
document.write( "2 -10   -8  \r\n" );
document.write( "3 -18   18  \r\n" );
document.write( "4   0   56  \r\n" );
document.write( "5  56  106  \r\n" );
document.write( "6 162 \r\n" );
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document.write( "They are not all the same number.  So,\r\n" );
document.write( "out to the right of that we do the same\r\n" );
document.write( "thing again.  That is, we subtract \r\n" );
document.write( "(-8)-(-22) = -8+22 = 14, then \r\n" );
document.write( "(18)-(-8) = 18+8 = 26, etc:\r\n" );
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document.write( "1  12  -22  14    \r\n" );
document.write( "2 -10   -8  26  \r\n" );
document.write( "3 -18   18  38  \r\n" );
document.write( "4   0   56  50\r\n" );
document.write( "5  56  106  \r\n" );
document.write( "6 162 \r\n" );
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document.write( "They are not all the same number.  So,\r\n" );
document.write( "we do the same thing again.  That is, \r\n" );
document.write( "we subtract \r\n" );
document.write( "(26)-(14) = 12, then \r\n" );
document.write( "(38)-(26) = 12, then\r\n" );
document.write( "(50)-(38) = 12 and we have\r\n" );
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document.write( "1  12  -22  14  12  \r\n" );
document.write( "2 -10   -8  26  12\r\n" );
document.write( "3 -18   18  38  12\r\n" );
document.write( "4   0   56  50\r\n" );
document.write( "5  56  106  \r\n" );
document.write( "6 162 \r\n" );
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document.write( "Now those are all the same number, 12.\r\n" );
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document.write( "It took 3 columns to get to a column that had \r\n" );
document.write( "all the same numbers in it.\r\n" );
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document.write( "That tells us we must use a 3rd degree polynomial\r\n" );
document.write( "equation, So we assume a 3rd degree polynomial equation:\r\n" );
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document.write( "y = Ax³ + Bx² + Cx + D\r\n" );
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document.write( "We substitute the first 4 points since\r\n" );
document.write( "we have 4 unknowns A,B,C, and D\r\n" );
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document.write( "y = Ax³ + Bx² + Cx + D\r\n" );
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document.write( "Substituting (1,12):\r\n" );
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document.write( "12 = A(1)³ + B(1)² + C(1) + D\r\n" );
document.write( "12 = A + B + C + D\r\n" );
document.write( "A + B + C + D = 12\r\n" );
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document.write( "Substituting (2,-10):\r\n" );
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document.write( "-10 = A(2)³ + B(2)² + C(2) + D\r\n" );
document.write( "-10 = 8A + 4B + 2C + D\r\n" );
document.write( "8A + 4B + 2C + D = -10\r\n" );
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document.write( "Substituting (3,-18):\r\n" );
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document.write( "-18 = A(3)³ + B(3)² + C(3) + D\r\n" );
document.write( "-18 = 27A + 9B + 3C + D\r\n" );
document.write( "27A + 9B + 3C + D = -18\r\n" );
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document.write( "Substituting (4,0):\r\n" );
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document.write( "0 = A(4)³ + B(4)² + C(4) + D\r\n" );
document.write( "0 = 64A + 16B + 4C + D\r\n" );
document.write( "64A + 16B + 4C + D = 0\r\n" );
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document.write( "So we have the system of 4 equations in 4 unknowns:\r\n" );
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document.write( "(1)    A +   B +  C + D =  12\r\n" );
document.write( "(2)   8A +  4B + 2C + D = -10\r\n" );
document.write( "(3)  27A +  9B + 3C + D = -18\r\n" );
document.write( "(4)  64A + 16B + 4C + D =   0\r\n" );
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document.write( "Subtract (1) from (2), (2) from (3) and (3) from (4)\r\n" );
document.write( "to eliminate D, and have 3 equations in 3 unknowns:\r\n" );
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document.write( "(5)  7A + 3B + C = -22\r\n" );
document.write( "(6) 19A + 5B + C =  -8\r\n" );
document.write( "(7) 37A + 7B + C =  18\r\n" );
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document.write( "Subtract (5) from (6) and (6) from (7)\r\n" );
document.write( "to eliminate C, and have 2 equations in 2 unknowns:\r\n" );
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document.write( "(8) 12A + 2B = 14\r\n" );
document.write( "(9) 18A + 2B = 26\r\n" );
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document.write( "Subtract (8) from (9)\r\n" );
document.write( "to eliminate B, and have 1 equation in 1 unknown:\r\n" );
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document.write( "(10) 6A = 12\r\n" );
document.write( "      A = 2\r\n" );
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document.write( "Substitute A = 2 in (8)\r\n" );
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document.write( "(8) 12(2) + 2B = 14\r\n" );
document.write( "       24 + 2B = 14\r\n" );
document.write( "            2B = -10\r\n" );
document.write( "             B = -5\r\n" );
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document.write( "Substitute A = 2 and B = -5 in (5)\r\n" );
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document.write( "(5)  7(2) + 3(-5) + C = -22\r\n" );
document.write( "         14 - 15 + C = -22\r\n" );
document.write( "              -1 + C = -22\r\n" );
document.write( "                   C = -21 \r\n" );
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document.write( "Substitute A = 2 and B = -5 and C = -21 in (1)\r\n" );
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document.write( "(1)    (2) + (-5) + (-21) + D = 12\r\n" );
document.write( "               2 - 5 - 21 + D = 12\r\n" );
document.write( "                      -24 + D = 12\r\n" );
document.write( "                            D = 36\r\n" );
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document.write( "So A = 2, B = -5, C = -21, D = 36\r\n" );
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document.write( "The desired polynomial\r\n" );
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document.write( "y = Ax³ + Bx² + Cx + D\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "y = 2x³ - 5x² - 21x + 36\r\n" );
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document.write( "The graph is \r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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