document.write( "Question 199686: solve the given polynomial equation, use rational zero theorem and Descrates rule of sighns as an aid in optaining the first root to begin synthetic division to give the solution set for 8x^3-10x^2-7x-1=0 The solution set is? \n" ); document.write( "
Algebra.Com's Answer #150096 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"8x%5E3-10x%5E2-7x-1=0\"\r\n" );
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document.write( "It has 1 sign change going from left to right.\r\n" );
document.write( "so it has 1 positive zero, which means it\r\n" );
document.write( "has either 2 or 0 negative zeros.\r\n" );
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document.write( "The only rational zeros it might have are the\r\n" );
document.write( "fractions whose numerators divide evenly into\r\n" );
document.write( "the last term and whose denominators divide\r\n" );
document.write( "evenly into the first coefficient, 8.\r\n" );
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document.write( "So the only possible rational zeros are\r\n" );
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document.write( "±1/1. ±1/2, ±1/4, ±1/8\r\n" );
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document.write( "or\r\n" );
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document.write( "±1. ±1/2, ±1/4, ±1/8\r\n" );
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document.write( "We try 1\r\n" );
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document.write( "1 | 8 -10 -7  -1\r\n" );
document.write( "  |     8 -2  -9 \r\n" );
document.write( "   -------------\r\n" );
document.write( "    8  -2 -9 -10\r\n" );
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document.write( "No luck.  We try 1/2\r\n" );
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document.write( "1/2 | 8 -10  -7 -1\r\n" );
document.write( "    |     4  -3 -5 \r\n" );
document.write( "     ---------------\r\n" );
document.write( "      8  -6 -10 -6\r\n" );
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document.write( "No luck.  We try 1/4\r\n" );
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document.write( "1/4 | 8 -10 -7    -1\r\n" );
document.write( "    |     2 -2  -9/4 \r\n" );
document.write( "     ---------------\r\n" );
document.write( "      8  -8 -9 -13/4\r\n" );
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document.write( "No luck.  We try 1/8\r\n" );
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document.write( "1/8 | 8 -10    -7      -1\r\n" );
document.write( "    |     1  -9/8  -65/64 \r\n" );
document.write( "     --------------------\r\n" );
document.write( "      8  -9 -65/8 -129/64\r\n" );
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document.write( "No luck.  We try -1\r\n" );
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document.write( "-1 | 8 -10 -7  -1\r\n" );
document.write( "   |    -8 18 -11 \r\n" );
document.write( "   --------------\r\n" );
document.write( "     8 -18 11 -12\r\n" );
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document.write( "No luck.  But, we learned that since the signs \r\n" );
document.write( "on that bottom row alternate and we are looking \r\n" );
document.write( "for a negative zero, that -1 is a lower bound\r\n" );
document.write( "for the negative zeros.  But heck, the only\r\n" );
document.write( "possibilities of a rational zero are greater \r\n" );
document.write( "than -1 anyway. So, that didn't really help.\r\n" );
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document.write( "We try -1/2\r\n" );
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document.write( "-1/2 | 8 -10 -7 -1\r\n" );
document.write( "     |    -4  7  0 \r\n" );
document.write( "     ---------------\r\n" );
document.write( "       8 -14  0 -1\r\n" );
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document.write( "No luck.  We try -1/4\r\n" );
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document.write( "-1/4 | 8 -10 -7 -1\r\n" );
document.write( "     |    -2  3  1 \r\n" );
document.write( "     --------------\r\n" );
document.write( "       8 -12 -4  0\r\n" );
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document.write( "Hey yeah! -1/4 is a zero\r\n" );
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document.write( "So now we have factored the original\r\n" );
document.write( "polynomial equation this way:\r\n" );
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document.write( "\"%28x%2B1%2F4%29%288x%5E2-12x-4%29=0\"\r\n" );
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document.write( "So now we factor the \"8x%5E2-12x-4\" by taking\r\n" );
document.write( "out 4. \r\n" );
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document.write( "\"%28x%2B1%2F4%294%282x%5E2-3x-1%29=0\"\r\n" );
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document.write( "\"4%28x%2B1%2F4%29%282x%5E2-3x-1%29=0\"\r\n" );
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document.write( "\"%284x%2B1%29%282x%5E2-3x-1%29=0\"\r\n" );
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document.write( "Setting each factor = 0:\r\n" );
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document.write( "\"4x%2B1=0\"\r\n" );
document.write( "  \"4x=-1\"\r\n" );
document.write( "   \"x=-1%2F4\"  we already knew that was one.\r\n" );
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document.write( "\"2x%5E2-3x-1=0\"\r\n" );
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document.write( "\"a+=+2\", \"b=-3\", \"c=-1\"\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+=+%283+%2B-+sqrt%289%2B8+%29%29%2F4+\"\r\n" );
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document.write( "\"x+=+%283+%2B-+sqrt%2817%29%29%2F4+\"\r\n" );
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document.write( "So there are three zeros:\r\n" );
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document.write( "\"x=-1%2F4\", \"x+=+%283+%2B+sqrt%2817%29%29%2F4+\", and \"x+=+%283+-+sqrt%2817%29%29%2F4+\"\r\n" );
document.write( " \r\n" );
document.write( "Edwin
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