document.write( "Question 1205602: Let f(x) = [(6+7x)/(2-x)(1+x^2)]
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\n" ); document.write( "f(x) = 3 + 5x - (1/2 x^2) - (15/4 x^3) \n" ); document.write( "

Algebra.Com's Answer #842600 by ikleyn(52776) $\"\"$ $\"About$

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\n" ); document.write( "Let f(x) = [(6+7x)/(2-x)(1+x^2)]
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document.write( "f(x) has this decomposition in partial fractions\r\n" );
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document.write( "     =  + .    (1)\r\n" );
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document.write( "with some real coefficients \"a\", \"b\" and \"c\".  \r\n" );
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document.write( "These coefficients are unknown now, and the rest part of this post is to determine them.\r\n" );
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document.write( "Multiply both sides of equality (1) by  . You will get\r\n" );
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document.write( "    6 + 7x = (ax+b)*(2-x) + .    (2)\r\n" );
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document.write( "At x= 2,  equation  (2)  gives\r\n" );
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document.write( "    6 + 7*2 = c*(1+2^2),  or  20 = 5c,  or  c= 20/5 = 4.\r\n" );
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document.write( "At x= 0 and c= 4,  equation  (2)  gives\r\n" );
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document.write( "    6 + 7*0 = (a*0+b)*(2-0) + 4*(1+0^2),  or  6 = 2b + 4;  then  6-4 = 2b,  2 = 2b,  b = 2/2 = 1.\r\n" );
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document.write( "At x= 1 and c= 4, b= 1,  equation  (2)  gives\r\n" );
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document.write( "    6 + 7*1 = (a*1+1)*(2-1) + 4*(1+1^2),  or  13 = (a+1) + 4*2;  then  13 - 8 = a+1;  5 = a+1;  a= 5-1 = 4.\r\n" );
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document.write( "ANSWER.   =  + .\r\n" );
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\n" ); document.write( "\n" ); document.write( "Part (1) is solved. \r
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