document.write( "Question 1116302: Find a polynomial function whose graph passes through (-1,-9),(0,-2),(1,1),(2,12) \n" ); document.write( "
Algebra.Com's Answer #731196 by Edwin McCravy(20055)\"\" \"About 
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document.write( "There are 4 points, so assume a general polynomial of\r\n" );
document.write( "degree which is one less, or a polynomial of degree 3.\r\n" );
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document.write( "Substitute each point (x,y) in Ax³ + Bx² + Cx + D = y:\r\n" );
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document.write( "A(-1)³ + B(-1)² + C(-1) + D = -9\r\n" );
document.write( " A(0)³ +  B(0)² +  C(0) + D = -2\r\n" );
document.write( " A(1)³ +  B(1)² +  C(1) + D =  1\r\n" );
document.write( " A(2)³ +  B(2)² +  C(2) + D = 12\r\n" );
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document.write( "Simplify and solve that system of 4 equations.\r\n" );
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document.write( "The second equation gives us D = -2, then the \r\n" );
document.write( "system simplifies to:\r\n" );
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document.write( "-A +  B -  C = -7\r\n" );
document.write( " A +  B +  C =  3\r\n" );
document.write( "8A + 4B + 2C = 14\r\n" );
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document.write( "Adding the first two equations gives\r\n" );
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document.write( "     2B      = -4\r\n" );
document.write( "      B      = -2\r\n" );
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document.write( "Substituting B = -2 in the 2nd and 3rd equations:\r\n" );
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document.write( " A +  (-2) +  C =  3\r\n" );
document.write( "8A + 4(-2) + 2C = 14\r\n" );
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document.write( "which simplifies to\r\n" );
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document.write( " A +  C =  5\r\n" );
document.write( "8A + 2C = 22\r\n" );
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document.write( "Multiplying the 1st by -2\r\n" );
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document.write( "-2A - 2C = -10\r\n" );
document.write( " 8A + 2C =  22\r\n" );
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document.write( " 6A      =  12\r\n" );
document.write( "  A      =   2\r\n" );
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document.write( "Substitute in \r\n" );
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document.write( " A +  C =  5\r\n" );
document.write( " 2 +  C =  5\r\n" );
document.write( "      C =  3   \r\n" );
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document.write( "So A=2, B=-2, C=3, D=-2 and\r\n" );
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document.write( "Ax³ + Bx² + Cx + D = y  becomes\r\n" );
document.write( "2x³ - 2x² + 3x - 2 = y\r\n" );
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document.write( "or use P(x) for y and write it on the left:\r\n" );
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document.write( "P(x) = 2x³ - 2x² + 3x - 2 = y\r\n" );
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document.write( "Here's the graph.  Notice that it passes through\r\n" );
document.write( "all four given points: \r\n" );
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document.write( "Edwin
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