document.write( "Question 119377: 12x^3-32x^2+25x-6=0 \n" ); document.write( "
Algebra.Com's Answer #87559 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "12x³ - 32x² + 25x - 6 = 0\r\n" );
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document.write( "It has three sign changes going from left to right,\r\n" );
document.write( "so it has 3 or 1 positive solutions.\r\n" );
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document.write( "f(-x) = 12(-x)³ - 32(-x)² + 25(-x) - 6 = -12x³ - 32x² - 25x - 6\r\n" );
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document.write( "has no sign changes so there are no negative solutions.\r\n" );
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document.write( "Any rational solutions it may have must have a numerator which\r\n" );
document.write( "is a factor of 6 and a denominator which is a factor of 12.\r\n" );
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document.write( "Possible numerators: 1,2,3,6\r\n" );
document.write( "Possible denominators: 1,2,3,4,6,12\r\n" );
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document.write( "Possible roots: 1/1, 1/2, 1/3, 1/4, 1/6, 1/12, 2/1, 2/2, 2/3, 2/6, \r\n" );
document.write( "2/12, 3/1, 3/2, 3/3, 3/4, 3/6, 3/12, 6/1, 6/2, 6/3, 6/4, 6/6, 6/12\r\n" );
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document.write( "Reducing the fractions: 1, 1/2, 1/3, 1/4, 1/6, 1/12, 2, 1, 2/3, 1/3, \r\n" );
document.write( "1/6, 3, 3/2, 1, 3/4, 1/2, 1/4, 6, 3, 2, 3/2, 1, 1/2\r\n" );
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document.write( "Eliminating the duplications: 1, 1/2, 1/3, 1/4, 1/6, 1/12, 2, 2/3,\r\n" );
document.write( "3, 3/2, 3/4, 1/4, 6\r\n" );
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document.write( "Try 1 as a solution by dividing synthetically\r\n" );
document.write( "by x-1\r\n" );
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document.write( "1 | 12 -32  25 -6\r\n" );
document.write( "  |     12 -20  5 \r\n" );
document.write( "    12 -20   5 -1\r\n" );
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document.write( "No it has 1 as a remainder, not 0, so x-1 is not\r\n" );
document.write( "a factor and therefore x=1 is not a solution\r\n" );
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document.write( "Try \"1%2F2\" as a zero by dividing synthetically\r\n" );
document.write( "by x-\"1%2F2\"\r\n" );
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document.write( "  \"1%2F2\"| 12 -32  25 -6\r\n" );
document.write( "    |      6 -13  6     \r\n" );
document.write( "      12 -26  12  0\r\n" );
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document.write( "This has a remainder of 0 so x-\"1%2F2\" is a factor and\r\n" );
document.write( "therefore x=\"1%2F2\" is a solution solution.  The \r\n" );
document.write( "quotient is 12x²-26+12.  That means that we have now\r\n" );
document.write( "factored the left side of \r\n" );
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document.write( "12x³ - 32x² + 25x - 6 = 0\r\n" );
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document.write( "as\r\n" );
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document.write( "(x - \"1%2F2\")(12x² - 26x - 6) = 0\r\n" );
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document.write( "Now we can factor 2 out of the the second parentheses:\r\n" );
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document.write( "(x - \"1%2F2\")2(6x² - 13x - 3) = 0\r\n" );
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document.write( "If you like you can multiply the 2 into the first\r\n" );
document.write( "parentheses:\r\n" );
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document.write( " (2x - 1)(6x² - 13x - 3) = 0\r\n" );
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document.write( "Now factor the second parentheses:\r\n" );
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document.write( "(2x - 1)(3x - 2)(2x - 3) = 0\r\n" );
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document.write( "Setting the first factor = 0, 2x - 1 = 0, gives solution x = \"1%2F2\"\r\n" );
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document.write( "Setting the second factor = 0, 3x - 2 = 0, gives solution x = \"2%2F3\"\r\n" );
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document.write( "Setting the third factor = 0, 2x - 3 = 0, gives solution x = \"3%2F2\"\r\n" );
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document.write( "Edwin
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