document.write( "Question 1049564: What are the sum of the coefficient of Q(x), when 37x^73-73x^37+36 is divided by x-1. \n" ); document.write( "

Algebra.Com's Answer #665155 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "What are the sum of the coefficient of Q(x), when 37x^73-73x^37+36 is divided by x-1.
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document.write( "     Let Q(x) be a quotient of division the polynomial P(x) =   by binomial (x-1).\r\n" );
document.write( "     Find the sum of coefficients of the polynomial Q(x).\r\n" );
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document.write( "1.  Notice that  x=1 is the root of the polynomial P(x) = .\r\n" );
document.write( "    It is easy to check directly:  P(1) = 0.\r\n" );
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document.write( "    Hence, the polynomial P(x) is divisible by the binomial (x-1) without a remainder, and the quotient  is a polynomial with integer coefficients.\r\n" );
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document.write( "    Again, the quotient Q(x) =  is a polynomial with integer coefficients, and the problem asks to find the sum of the coefficients of this polynomial.\r\n" );
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document.write( "2.  For any polynomial F(x), the sum of its coefficients is equal to the value of the polynomial at x=1:\r\n" );
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document.write( "    the sum of coefficients of F(x) is equal to F(1).\r\n" );
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document.write( "3.  Based on it, the first idea is to calculate Q(1) by substituting directly x=1 into the quotient .\r\n" );
document.write( "    But it doesn't work, since the denominator becomes 0 (zero) at x=1.\r\n" );
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document.write( "4.  Although this crude idea doesn't work, there is another way to calculate Q(1) as the limit of Q(x) at x-->1.\r\n" );
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document.write( "    Notice that Q(x) =  ==  = , \r\n" );
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document.write( "    therefore  Q(1) = limit of  at x-->1  = (derivative of P(x) at x=1)  =  (P'(x) at x=1) = P'(1).\r\n" );
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document.write( "    P'(x) is easy to express. It is :  P'(x) = \r\n" );
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document.write( "    and therefore P'(1) = 37*73 - 73*37 = 0.\r\n" );
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document.write( "5.  Thus we have this chain of equalities:\r\n" );
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document.write( "    sum of coefficients Q(x) = Q(1) = (lim Q(x) at x-->1)  = P'(1) = 0.\r\n" );
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document.write( "Answer.  The sum of coefficients of the quotient  is 0 (zero).\r\n" );
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