document.write( "Question 334192: I really need help with this problem: Consider 2x^4+11x^3+x^2-10x-4. Divide by
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Algebra.Com's Answer #239472 by Edwin McCravy(20062)\"\" \"About 
You can put this solution on YOUR website!
\"2x%5E4%2B11x%5E3%2Bx%5E2-10x-4\"
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document.write( "If it has any rational solutions the numerator must be\r\n" );
document.write( "a factor of 4 and the denominator must be a factor of 2,\r\n" );
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document.write( "The only possible rational solutions are \r\n" );
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document.write( "\"%22%22+%2B-+1\", \"%22%22+%2B-+2\", \"%22%22+%2B-+4\", or \"%22%22+%2B-+1%2F2\"\r\n" );
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document.write( "Trying the easiest one x = 1\r\n" );
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document.write( "So we divide by x - 1, either synthetically:\r\n" );
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document.write( "1|2  11   1  -10  -4\r\n" );
document.write( " |    2  13   14   4 \r\n" );
document.write( "  2  13  14    4   0\r\n" );
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document.write( "or by long division   \r\n" );
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document.write( "             2x³ + 13x² + 14x + 4\r\n" );
document.write( "x - 1)2x4 + 11x³ +   x² - 10x - 4 \r\n" );
document.write( "      2x4 -  2x³\r\n" );
document.write( "            13x³ +   x²\r\n" );
document.write( "            13x³ - 13x²\r\n" );
document.write( "                   14x² - 10x\r\n" );
document.write( "                   14x² - 14x\r\n" );
document.write( "                           4x - 4\r\n" );
document.write( "                           4x - 4\r\n" );
document.write( "                                0     \r\n" );
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document.write( "You have now factored the original polynomial as\r\n" );
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document.write( "(x - 1)(2x³ + 13x² + 14x + 4)\r\n" );
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document.write( "Next let's do the same with the cubic polynomial in\r\n" );
document.write( "the second parenthesis:\r\n" );
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document.write( "If it has any rational solutions the numerator must be\r\n" );
document.write( "a factor of 4 and the denominator must be a factor of 2,\r\n" );
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document.write( "We've already done that. The only possible rational solutions are \r\n" );
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document.write( "\"%22%22+%2B-+1\", \"%22%22+%2B-+2\", \"%22%22+%2B-+4\", or \"%22%22+%2B-+1%2F2\"\r\n" );
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document.write( "You could go to the trouble of trying all those.  But sooner of\r\n" );
document.write( "later you'd get around to trying to divide by \"-1%2F2\" or \"-.5\"\r\n" );
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document.write( "Divide by x + .5, either synthetically:\r\n" );
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document.write( "-.5|2  13  14   4\r\n" );
document.write( "   |   -1  -6  -4    \r\n" );
document.write( "    2  12   8   0\r\n" );
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document.write( "or by long division   \r\n" );
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document.write( "               2x² + 12x + 8 \r\n" );
document.write( "x + .5)2x³ + 13x² + 14x + 4 \r\n" );
document.write( "       2x³ +   x²\r\n" );
document.write( "             12x² + 14x\r\n" );
document.write( "             12x? +  6x\r\n" );
document.write( "                     8x + 4\r\n" );
document.write( "                     8x + 4\r\n" );
document.write( "                          0         \r\n" );
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document.write( "And you have now factored further as\r\n" );
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document.write( "(x - 1)(x + .5)(2x² + 12x + 8)\r\n" );
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document.write( "Now we can factor 2 out of the third parentheses:\r\n" );
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document.write( "(x - 1)(x + .5)2(x² + 6x + 4)\r\n" );
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document.write( "2(x - 1)(x + .5)(x² + 6x + 4)\r\n" );
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document.write( "The quadratic will not factor but we can find its roots \r\n" );
document.write( "by using the quadratic formula:\r\n" );
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document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"\r\n" );
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document.write( "\"x+=+%28-6+%2B-+sqrt%28+6%5E2-4%2A1%2A4+%29%29%2F%282%2A1%29+\"\r\n" );
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document.write( "\"x+=+%28-6+%2B-+sqrt%2836-16+%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%28-6+%2B-+sqrt%2820%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%28-6+%2B-+sqrt%284%2A5%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%28-6+%2B-+2sqrt%285%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%282%28-3+%2B-+sqrt%285%29%29%29%2F2+\"\r\n" );
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document.write( "\"x+=+%28cross%282%29%28-3+%2B-+sqrt%285%29%29%29%2Fcross%282%29+\"\r\n" );
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document.write( "\"x+=+-3+%2B-+sqrt%285%29+\"\r\n" );
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document.write( "So the 4 roots are\r\n" );
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document.write( "1, \"-1%2F2\", \"-3+%2B+sqrt%285%29+\", and \"-3+-+sqrt%285%29+\"\r\n" );
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document.write( "Edwin
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