document.write( "Question 1051131: Find the smallest positive integer and the largest
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\n" ); document.write( "P(x) = 2x^3 + x^2 - 25x + 9 \n" ); document.write( "
Algebra.Com's Answer #666731 by Edwin McCravy(20055)\"\" \"About 
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Find the smallest positive integer and the largest
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\n" ); document.write( "P(x) = 2x^3 + x^2 - 25x + 9
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document.write( "If we do synthetic division with a\r\n" );
document.write( "positive number and find that all\r\n" );
document.write( "the signs on the bottom row of the\r\n" );
document.write( "synthetic division are non-negative,\r\n" );
document.write( "then the positive number that we are\r\n" );
document.write( "doing the synthetic division with is\r\n" );
document.write( "an upper bound for all the real zeros.\r\n" );
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document.write( "We try 1 for an upper bound:\r\n" );
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document.write( "1 | 2 1 -25   9\r\n" );
document.write( "  |   2   3 -22\r\n" );
document.write( "    2 3 -22 -13\r\n" );
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document.write( "No the bottom row of numbers are not\r\n" );
document.write( "all non-negative.\r\n" );
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document.write( "We try 2 for an upper bound:\r\n" );
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document.write( "2 | 2 1 -25   9\r\n" );
document.write( "  |   4  10 -30\r\n" );
document.write( "    2 5 -15 -21\r\n" );
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document.write( "No the bottom row of numbers are not\r\n" );
document.write( "all non-negative. \r\n" );
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document.write( "We try 3 for an upper bound:\r\n" );
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document.write( "3 | 2 1 -25   9\r\n" );
document.write( "  |   6  21 -12\r\n" );
document.write( "    2 7  -4  -3\r\n" );
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document.write( "No the bottom row of numbers are not\r\n" );
document.write( "all non-negative.\r\n" );
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document.write( "We try 4 for an upper bound:\r\n" );
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document.write( "4 | 2 1 -25  9\r\n" );
document.write( "  |   8  36 44\r\n" );
document.write( "    2 9  11 53\r\n" );
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document.write( "Yes the bottom row of numbers are all \r\n" );
document.write( "non-negative.\r\n" );
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document.write( "Therefore 4 is the smallest positive integer that, \r\n" );
document.write( "by the Upper- and Lower-Bound Theorem, is an upper \r\n" );
document.write( "bound for the real zeros of the polynomial function.\r\n" );
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document.write( "If we do synthetic division with a\r\n" );
document.write( "negative number and find that the \r\n" );
document.write( "signs on the bottom row of the\r\n" );
document.write( "synthetic division alternate in sign,\r\n" );
document.write( "then the negative number that we are\r\n" );
document.write( "doing the synthetic division with is\r\n" );
document.write( "a lower bound for all the real zeros.\r\n" );
document.write( "[In this process we can consider 0\r\n" );
document.write( "on the bottom row to have either sign].\r\n" );
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document.write( "We try -1 for a lower bound:\r\n" );
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document.write( "-1 | 2  1 -25   9\r\n" );
document.write( "   |   -1   0  25\r\n" );
document.write( "     2  0 -25  33\r\n" );
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document.write( "No the bottom row of numbers do not\r\n" );
document.write( "alternate in sign.\r\n" );
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document.write( "We try -2 for a lower bound:\r\n" );
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document.write( "-2 | 2  1 -25   9\r\n" );
document.write( "   |   -4   6  38\r\n" );
document.write( "     2 -3 -19  47\r\n" );
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document.write( "No the bottom row of numbers do not\r\n" );
document.write( "alternate in sign.\r\n" );
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document.write( "We try -3 for a lower bound:\r\n" );
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document.write( "-3 | 2  1 -25   9\r\n" );
document.write( "   |   -6  15  30\r\n" );
document.write( "     2 -5 -10  39\r\n" );
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document.write( "No the bottom row of numbers do not\r\n" );
document.write( "alternate in sign.\r\n" );
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document.write( "We try -4 for a lower bound:\r\n" );
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document.write( "-4 | 2  1 -25   9\r\n" );
document.write( "   |   -8  28 -12\r\n" );
document.write( "     2 -7   3  -3\r\n" );
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document.write( "Yes the bottom row of numbers alternate \r\n" );
document.write( "in sign, so \r\n" );
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document.write( "Therefore -4 is the largest negative integer that, by \r\n" );
document.write( "the Upper- and Lower-Bound Theorem, is a lower bound \r\n" );
document.write( "for the real zeros of the polynomial function.\r\n" );
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document.write( "Edwin
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