document.write( "Question 762287: Find the lowest degree polynomial f(x) that match the graph below. Leave your answer in factored form.\r
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Algebra.Com's Answer #463839 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "1. The graph has 4 turning points, so the lowest degree it can have\r\n" );
document.write( "   is degree which is 1 more than the number of turning points 5.  \r\n" );
document.write( "   So it has degree 5.\r\n" );
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document.write( "2. The graph touches and \"bounces off\" the x-axis at (-6,0) and (5,0),\r\n" );
document.write( "   so x=-6 and x=5 are zeros of even multiplicity.  The least possible even\r\n" );
document.write( "   multiplicity is 2.  Therefore f(x) has factors (x+6)² and (x-5)² \r\n" );
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document.write( "3. The graph cuts through the x-axis at (2,0), So x=2 is a zero of odd\r\n" );
document.write( "   multiplicity.  The least possible odd multiplicity is 1.  Therefore,\r\n" );
document.write( "   f(x) has factor (x-2).\r\n" );
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document.write( "4. f(x) contains the factors (x+6)²(x-5)²(x-2). That would multiply out\r\n" );
document.write( "   to be a fifth degree polynomial but it may also have a constant factor\r\n" );
document.write( "   other than 1 as well.  We will let the contant factor be k.  \r\n" );
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document.write( "So we know that f(x) has this form:\r\n" );
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document.write( "         f(x) = k(x+6)²(x-5)²(x-2). \r\n" );
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document.write( "We only need to determine the value of k.  We do that by observing that \r\n" );
document.write( "the graph has y-intercept (0,4).  Therefore f(0) = 4.  So we substitute\r\n" );
document.write( "0 for x in f(x) and set it equal to 4:\r\n" );
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document.write( "         f(0) = k(0+6)²(0-5)²(0-2) = 4 =\r\n" );
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document.write( "                   k(-6)²(-5)²(-2) = 4 \r\n" );
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document.write( "                     k(36)(25)(-2) = 4\r\n" );
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document.write( "                            -1800k = 4\r\n" );
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document.write( "                                 k = \"4%2F%28-1800%29\"\r\n" );
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document.write( "                                 k = \"-1%2F450\"\r\n" );
document.write( "Therefore:\r\n" );
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document.write( "           f(x) = k(x+6)²(x-5)²(x-2)\r\n" );
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document.write( "becomes    \r\n" );
document.write( "           f(x) = \"-1%2F450\"(x+6)²(x-5)²(x-2).\r\n" );
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document.write( "Here is a more accurate graph than the one above:\r\n" );
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document.write( "Edwin
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