document.write( "Question 704183: the zeroes of a polynomial function are 1,1,-5/2 and -1, develope the function and then sketch the graph. \n" ); document.write( "

Algebra.Com's Answer #433969 by AnlytcPhil(1807) $\"\"$ $\"About$

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document.write( "First we'll develop a polynomial equation with\r\n" );
document.write( "solutions 1,1,-5/2, and -1.\r\n" );
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document.write( "  x = 1,    x = 1,      x = -5/2,    x = -1\r\n" );
document.write( "x-1 = 0   x-1 = 0   x+5/2 = 0      x+1 = 0\r\n" );
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document.write( "         (x-1)(x-1)(x+5/2)(x+1) = 0\r\n" );
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document.write( "To avoid fractions multiply through by 2\r\n" );
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document.write( "      (x-1)(x-1)(2)(x+5/2)(x+1) = 0\r\n" );
document.write( "          (x-1)(x-1)(2x+5)(x+1) = 0\r\n" );
document.write( "            (x²-2x+1)(2x²+7x+1) = 0\r\n" );
document.write( "               2x⁴+3x³-7x²-3x+5 = 0\r\n" );
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document.write( "That's a quartic equation with those roots.\r\n" );
document.write( "Therefore a quartic function with those zeros is:\r\n" );
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document.write( "   p(x) = 2x⁴+3x³-7x²-3x+5\r\n" );
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document.write( "To graph it\r\n" );
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document.write( "1. the coefficient of the leading term 2x⁴ is positive\r\n" );
document.write( "and therefor the graph goes upward on the extreme right side.\r\n" );
document.write( "2. the degree is 4, which is an even number, so the graph\r\n" );
document.write( "also goes up on the extreme left. \r\n" );
document.write( "The first zero on the left is -5/2 or -2.5, so the curve\r\n" );
document.write( "comes downard from the left and since -2.5 has odd multiplicity,\r\n" );
document.write( "1, it cuts through the x-axis at -2.5 going downward to the right.\r\n" );
document.write( "2. The next zero is -1, so the graph must turn around and go\r\n" );
document.write( "upward.  Since -1 has odd multiplicity, 1, it cuts through the \r\n" );
document.write( "x-axis at -1 going upward to the right.\r\n" );
document.write( "3. The next and final zero is 1, so the graph must turn around and go\r\n" );
document.write( "downward.  Since 1 has even multiplicity, 2, the graph bounces off \r\n" );
document.write( "the x-axis at 1 going back upward to the right. \r\n" );
document.write( "You might plot these points:\r\n" );
document.write( "(-2.6,4.1), (-2.5,0), (-2,-9), (-1.3,-3.8), (-1,0), (0,5), (.5,2.25),\r\n" );
document.write( "(1,0),(1.5,5)    \r\n" );
document.write( "   \r\n" );
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document.write( "Edwin

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