document.write( "Question 1117810: What is the value of the constant term of \"%282x%5E3-1%2Fx%5E2%29%5E10\" \n" ); document.write( "
Algebra.Com's Answer #732979 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Writing by other tutors leaves you without a guidance on how to get this magic/miracle value  k = 6  (the \"seventh term\").\r
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\n" ); document.write( "\n" ); document.write( "            Therefore,  below I placed the full solution with detailed explanations of all auxiliary moments.\r
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document.write( "The binomial expansion is this formula\r\n" );
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document.write( "\"%28a%2Bb%29%5En\" = \"a%5En\" + \"C%5Bn%5D%5E1%2Aa%5E%28n-1%29%2Ab\" + \"C%5Bn%5D%5E2%2Aa%5E%28n-2%29%2Ab%5E2\" + \"C%5Bn%5D%5E3%2Aa%5E%28n-3%29%2Ab%5E3\" + . . . + \"C%5Bn%5D%5E%28n-1%29%2Aa%5E1%2Ab%5E%28n-1%29\" + \"b%5En\"\r\n" );
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document.write( "In our case,  n = 10,  a = \"2x%5E3\",  b = \"-1%2Fx%5E2\",  therefore, the binomial expansion in out case is  \r\n" );
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document.write( "\"%282x%5E3-+1%2Fx%5E2%29%5E10\" = \"%282x3%29%5E10\" + \"C%5B10%5D%5E1%2A%282x%5E3%29%5E9%2A%28-1%2Fx%5E2%29\" + \"C%5B10%5D%5E2%2A%282x%5E3%29%5E8%2A%28-1%2Fx%5E2%29%5E2\" + \"C%5B10%5D%5E3%2A%282x%5E3%29%5E7%2A%28-1%2Fx%5E2%29%5E3\" + . . . + \"C%5B10%5D%5E9%2A%282x%5E3%29%2A%28-1%2Fx%5E2%29%5E9\" + \"%28-1%2Fx%5E2%29%5E10\".\r\n" );
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document.write( "The common term is  \"C%5B10%5D%5Ek%2A%282x%5E3%29%5E%2810-k%29%2A%28-1%2Fx%5E2%29%5Ek\".\r\n" );
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document.write( "This common term is the constant term  at  3*(10-k) - 2k = 0,  i.e.  30 - 3k - 2k =0,  or  30 = 5k  ====>  k = \"30%2F5\" = 6 (the seventh term).\r\n" );
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document.write( "Then the coefficient at this term is  \"C%5B10%5D%5E6%2A2%5E%2810-6%29%2A%28-1%29%5E6\" = \"C%5B10%5D%5E6%2A2%5E4%2A%28-1%29%5E6\" = \"%28%2810%2A9%2A8%2A7%29%2F%281%2A2%2A3%2A4%2A5%2A6%29%29%2A2%5E4\" = 3360.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problems like this one are very standard,  and you can often meet them on exams or on tests.\r
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\n" ); document.write( "\n" ); document.write( "Therefore,  it is beneficiary for you to have and to know this standard and full solution.\r
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\n" ); document.write( "\n" ); document.write( "             * * * Memorize it as a mantra ! * * * \r
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\n" ); document.write( "\n" ); document.write( "To see other similar solved problems,  look into the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion,  Problem  2\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on binomial coefficients,  Problems  1  and  2\r
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