document.write( "Question 37138This question is from textbook
\n" ); document.write( ": Find a polynomial of degree 3 with roots 2i, 1, and 3.\r
\n" ); document.write( "\n" ); document.write( "Find the rational roots if they exist of x3-3x2-2x-8\r
\n" ); document.write( "\n" ); document.write( "Determine the positive and negative real roots of 2x5+x4-3x3+x2-x+1\r
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Algebra.Com's Answer #22835 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
Find the rational roots if they exist of\r\n" );
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document.write( " x³ - 3x² - 2x - 8 \r\n" );
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document.write( "If there are any rational roots, they will be fractions whose\r\n" );
document.write( "numerator divides evenly into 8, the constant term, and whose \r\n" );
document.write( "denominator divides evenly into 1, the coefficient of the \r\n" );
document.write( "highest power of x. They can also be positive or negative.  \r\n" );
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document.write( "Possible numerators are the factor of 8, i.e., 1, 2, 4, and 8 \r\n" );
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document.write( "Possible denominator is the only factor of 1, namely 1. So\r\n" );
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document.write( "Possible rational roots are ±1/1, ±2/1, ±4/1, and ±8/1 or\r\n" );
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document.write( "      ±1, ±2, ±4, and ±8\r\n" );
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document.write( "Try x=1, using synthetic division\r\n" );
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document.write( "      1| 1  -3  -2   -8\r\n" );
document.write( "       |     1  -2   -4       \r\n" );
document.write( "         1  -2  -4  -12\r\n" );
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document.write( "No, for we don't get 0 remainder\r\n" );
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document.write( "So we try x=-1, using synthetic division\r\n" );
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document.write( "     -1| 1  -3  -2   -8\r\n" );
document.write( "       |    -1   4   -2       \r\n" );
document.write( "         1  -4   2  -10\r\n" );
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document.write( "No, for we don't get 0 remainder\r\n" );
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document.write( "So we try x=2, using synthetic division\r\n" );
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document.write( "      2| 1  -3  -2   -8\r\n" );
document.write( "       |     2  -2   -8       \r\n" );
document.write( "         1  -1  -4  -16\r\n" );
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document.write( "No, for we don't get 0 remainder\r\n" );
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document.write( "So we try x=-2, using synthetic division\r\n" );
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document.write( "     -2| 1  -3  -2   -8\r\n" );
document.write( "       |    -2  10  -16       \r\n" );
document.write( "         1  -5   8  -24\r\n" );
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document.write( "No, for we don't get 0 remainder\r\n" );
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document.write( "So we try x=4, using synthetic division\r\n" );
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document.write( "      4| 1  -3  -2   -8\r\n" );
document.write( "       |     4   4    8       \r\n" );
document.write( "         1   1   2    0\r\n" );
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document.write( "Hey that gives 0 remainder!  So we have now factored \r\n" );
document.write( "the polynomial as\r\n" );
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document.write( "      (x - 4)(1x² + 1x + 2)\r\n" );
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document.write( "or\r\n" );
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document.write( "      (x - 4)(x² + x + 2)\r\n" );
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document.write( "Since the trinomial in the second parentheses doesn't\r\n" );
document.write( "factor, 4 is the only rational root. \r\n" );
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document.write( "Determine the number of positive and negative real roots of\r\n" );
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document.write( "2x5 + x4 - 3x3 + x2 - x + 1 \r\n" );
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document.write( "To find the number of positive real roots, we use DesCartes'\r\n" );
document.write( "rule of signs.  Count the sign changes going left to right\r\n" );
document.write( "when the polynomial is in descending order:\r\n" );
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document.write( "2x5 has sign + and x4 also has sign +, so that's NOT a sign change.\r\n" );
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document.write( "x4 has sign + and -3x3 has sign -, that's a change in sign, \r\n" );
document.write( "so that makes 1 sign change so far.\r\n" );
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document.write( "-3x3 has sign - and +x2 has sign +, that's a change in sign, \r\n" );
document.write( "so that makes 2 sign changes so far.\r\n" );
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document.write( "+x2 has sign + and -x has sign -, that's a change in sign, \r\n" );
document.write( "so that makes 3 sign changes so far.\r\n" );
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document.write( "-x has sign - and +1 has sign +, that's a change in sign, \r\n" );
document.write( "so that makes 4 sign changes so far.\r\n" );
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document.write( "So there are either 4 positive roots, 2 positive roots, or\r\n" );
document.write( "0 positive root.\r\n" );
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document.write( "To find the possible number of negative real roots, we replace x by\r\n" );
document.write( "-x, simplify, and find the number of positive roots of the resulting\r\n" );
document.write( "polynomial.\r\n" );
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document.write( "2(-x)5 + (-x)4 - 3(-x)3 + (-x)2 - (-x) + 1\r\n" );
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document.write( "-2x5 + x4 + 3x3 + x2 + x + 1\r\n" );
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document.write( "We can easily see there is only one sign change here, so\r\n" );
document.write( "there is exactly one negative root.\r\n" );
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document.write( "So there are either 4 or 2 or 1 positive roots, and exactly 1 negative \r\n" );
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document.write( "Find the other roots of x3+4x2-7x-10 if one root is 2\r\n" );
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document.write( "Use synthetic division:\r\n" );
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document.write( "        2|1  4 -7 -10\r\n" );
document.write( "         |   2 12  10\r\n" );
document.write( "          1  6  5   0\r\n" );
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document.write( "So the polynomial is now factored as\r\n" );
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document.write( "      (x - 2)(x2 + 6x + 5)\r\n" );
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document.write( "Now factor the trinomial in parentheses\r\n" );
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document.write( "     (x - 2)(x + 1)(x + 5)\r\n" );
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document.write( "Set each factor = 0, and solve for x and we get\r\n" );
document.write( "that the zeros are 2, -1, and -5\r\n" );
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document.write( "The other roots besides 2 are -1 and -5.\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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