document.write( "Question 237265: use descartes rule of signs to determine the possible number of positive real zeros and negative zeros. P(x)=-3x^4+2x^3-4x^2+x-11 \n" ); document.write( "

Algebra.Com's Answer #174481 by Edwin McCravy(20055) $\"\"$ $\"About$

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use descartes rule of signs to determine the possible number of positive real zeros and negative zeros. $\"P%28x%29=-3x%5E4%2B2x%5E3-4x%5E2%2Bx-11\"$
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document.write( "List all the terms of \r\n" );
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document.write( "Term #1. . Therefore, term #1 has a NEGATIVE sign\r\n" );
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document.write( "Term #2. . Therefore, term #2 has a POSITIVE sign\r\n" );
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document.write( "Therefore in going from Term #1 to Term #2 there was a change in \r\n" );
document.write( "sign, from negative to positive.  So far that's 1 sign change.\r\n" );
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document.write( "Term #3.  Therefore, term #2 has a NEGATIVE sign\r\n" );
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document.write( "Therefore in going from Term #2 to Term #3 there was another change \r\n" );
document.write( "in sign, from positive to negative.  So far that's 2 sign changes.\r\n" );
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document.write( "Term #4. . Therefore, term #2 has a POSITIVE sign\r\n" );
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document.write( "Therefore in going from Term #3 to Term #4 there was another change in \r\n" );
document.write( "sign, from negative to positive.  So far that's 3 sign change.\r\n" );
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document.write( "Term #5. . Therefore, term #5 has a NEGATIVE sign\r\n" );
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document.write( "Therefore in going from Term #4 to Term #5 there was another change \r\n" );
document.write( "in sign, from positive to negative.  So far that's 4 sign changes.\r\n" );
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document.write( "So the fact that there are 4 sign changes means that the maximum number of\r\n" );
document.write( "positive zeros is 4.  But since complex imaginary zeros come in pairs,\r\n" );
document.write( "There may be only 2 positive zeros, (in case there happens to be one pair\r\n" );
document.write( "of complex imaginary zeros).  Also there may be 0 positive zeros, (in case\r\n" );
document.write( "there happens to be two pairs of complex imaginary zeros).\r\n" );
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document.write( "So there are either 4 or 2 or 0 positive 0's.\r\n" );
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document.write( "Now the number negative zeros of  will be the number of\r\n" );
document.write( "positive zeros of  \r\n" );
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document.write( "So let's substitute  for  in  to get\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "List all the terms of \r\n" );
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document.write( "Term #1.  .  Therefore Term #1 has a NEGATIVE sign.\r\n" );
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document.write( "Term #2.  .  Therefore Term #2 also has a NEGATIVE sign. \r\n" );
document.write( "Therefore there is no sign change from Term #1 to Term #2.  So far 0 sign changes.\r\n" );
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document.write( "Term #3.  .  Therefore Term #3 also has a NEGATIVE sign. \r\n" );
document.write( "Therefore there is no sign change from Term #2 to Term #3.  So far there\r\n" );
document.write( "are still 0 sign changes.\r\n" );
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document.write( "Term #4.  .  Therefore Term #4 also has a NEGATIVE sign. \r\n" );
document.write( "Therefore there is no sign change from Term #3 to Term #4.  So far there\r\n" );
document.write( "are still 0 sign changes.\r\n" );
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document.write( "Term #5.  .  Therefore Term #5 also has a NEGATIVE sign. \r\n" );
document.write( "Therefore there is no sign change from Term #4 to Term #5.  So far there\r\n" );
document.write( "are still 0 sign changes.  \r\n" );
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document.write( "So there are 0 positive zeros of  and therefore there are\r\n" );
document.write( "0 negative zeros of P(x).\r\n" );
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document.write( "Answer: P(x) has 4 or 2 or 0 POSITIVE zeros and 0 NEGATIVE zeros. \r\n" );
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document.write( "[By doing further study on P(x), which you will eventually learn to\r\n" );
document.write( "do, we could discover that P(x) actually has 0 real zeros, and 2 pairs\r\n" );
document.write( "of complex imaginary zeros.]\r\n" );
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document.write( "Edwin

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