Algebra.Com's Answer #112387 by Edwin McCravy(20060)![\"\"](\"/images/stars/stars-13.gif\")  
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Consider polynomial  ; what is its maximum number of zeroes? @ The maximum number of zeroes = n, regardless of the value of n.
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document.write( "a. 0
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document.write( "b. 1
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document.write( "c. Cannot be determined without first knowing the value of n
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document.write( "d. None of the above
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document.write( "We apply Descartes' rule of signs, which says:\r\n" );
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document.write( "If f(x) = a polynomial in descending order, then:\r\n" );
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document.write( "1. The maximum number of positive zeros of polynomial f(x),\r\n" );
document.write( "is the number of sign changes in f(x), going left to right.\r\n" );
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document.write( "2. The maximum number of negative zeros of polynomial f(x),\r\n" );
document.write( "is the number of sign changes in f(-x), going left to right.\r\n" );
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document.write( "has only two terms, and NO sign changes. Therefore it has NO\r\n" );
document.write( "POSITIVE zeros.\r\n" );
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document.write( "Now let's test it for NEGATIVE zeros.\r\n" );
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document.write( "We form  by replacing  by  .\r\n" );
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document.write( "Now since we do not know whether n is even or odd, we have to\r\n" );
document.write( "consider both cases:\r\n" );
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document.write( "If  is even, then  is odd, so  and\r\n" );
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document.write( "So in this case\r\n" );
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document.write( "That has 1 sign change, so g(x) would have a maximum of 1 negative zero.\r\n" );
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document.write( "If is odd, then is even, so  and\r\n" );
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document.write( "So in this case\r\n" );
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document.write( "That also has 1 sign change, so g(x) would have a maximum of 1 \r\n" );
document.write( "negative zero regardless of whether n is even or odd.\r\n" );
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document.write( "Now we know that it has no positive zeros, a maximium of\r\n" );
document.write( "1 negative zero.\r\n" );
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document.write( "However we must also check to see if 0, which is neither\r\n" );
document.write( "positive nor negative, is a zero. So we find  \r\n" );
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document.write( "Therefore 0 is also a zero, regardless of n.\r\n" );
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document.write( "So the maximium number of zeros is 2. \r\n" );
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document.write( "Since 2 is not listed as a choice, then the only correct\r\n" );
document.write( "choice is d \"None of the above\".\r\n" );
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document.write( "Comment:\r\n" );
document.write( "It's easy to see the correct choice cannot be 0 or 1 \r\n" );
document.write( "because when  \r\n" );
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document.write( "This has two zeros 0 and -4.\r\n" );
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document.write( "And we showed above that c cannot be the answer,\r\n" );
document.write( "for we were able to determine that the maximum number\r\n" );
document.write( "of zeros is 2. \r\n" );
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document.write( "Edwin
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