document.write( "Question 41477: P(x)=x^5+3x^3+2x-6 \n" ); document.write( "
Algebra.Com's Answer #26765 by AnlytcPhil(1810)\"\" \"About 
You can put this solution on YOUR website!
You posted no instructions at all!!!!!\r\n" );
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document.write( "So I'll just have to guess what they might have been:\r\n" );
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document.write( "My guesses:\r\n" );
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document.write( "Use DesCartes rule of signs to determine the number of positive\r\n" );
document.write( "and negative zeros of P(x). \r\n" );
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document.write( "Find all real zeros of P(x).\r\n" );
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document.write( "If neither of these were the instructions, post again and\r\n" );
document.write( "this time give the instructions as to what you are supposed to do\r\n" );
document.write( "with P(x)\r\n" );
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document.write( "P(x) = x5 + 3x3 + 2x - 6\r\n" );
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document.write( "x5 and 3x3 are both positive, so there is no sign change going\r\n" );
document.write( "from x5 to 3x3.\r\n" );
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document.write( "3x3 and 2x are both positive, so there is no sign change going\r\n" );
document.write( "from 3x3 to 2x.\r\n" );
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document.write( "2x is positive and -6 is negative, so that is 1 sign change\r\n" );
document.write( "going from 2x to -6.\r\n" );
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document.write( "So there is exactly 1 positive zero of P(x).\r\n" );
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document.write( "Now we form P(-x)\r\n" );
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document.write( "P(-x) = (-x)5 + 3(-x)3 + 2(-x) - 6\r\n" );
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document.write( "P(-x) = -x5 - 3x3 - 2x - 6\r\n" );
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document.write( "-x5 and -3x3 are both negative, so there is no sign change going\r\n" );
document.write( "from -x5 to -3x3.\r\n" );
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document.write( "-3x3 and -2x are both positive, so there is no sign change going\r\n" );
document.write( "from -3x3 to -2x.\r\n" );
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document.write( "-2x and -6 are both negative, so there is no sign change\r\n" );
document.write( "going from -2x to -6.\r\n" );
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document.write( "So there are no negative zeros of P(x).\r\n" );
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document.write( "There is only one positive zero.\r\n" );
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document.write( "Since the leading coefficient is 1, the possible rational \r\n" );
document.write( "zeros are the ± divisors of the absolute value of the last\r\n" );
document.write( "term, |-6| or 6.  So we try the smallest divisor of 6, which\r\n" );
document.write( "is 1. \r\n" );
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document.write( "For synthetic division, we must insert placeholders for the\r\n" );
document.write( "powers of x which are missing in P(x)\r\n" );
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document.write( "P(x) = x5 + 0x4 + 3x3 + 0x2 + 2x - 6\r\n" );
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document.write( "1|1  0  3  0  2 -6\r\n" );
document.write( " |   1  1  4  4  6\r\n" );
document.write( "  1  1  4  4  6  0\r\n" );
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document.write( "Yes that is a zero since the last number on the bottom is 0.\r\n" );
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document.write( "So 1 is the only real zero of P(x).\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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