document.write( "Question 854282: Let f(x)=(2x^2-7x-1)/(x-2)(x^2+3).
\n" ); document.write( "(i)Express f(x) in partial fractions.
\n" ); document.write( "(ii)Hence obtain the expression of f(x) in ascending powers of x, up to and including the term x^2.
\n" ); document.write( "Please Help! \n" ); document.write( "
Algebra.Com's Answer #514603 by Vladdroid(91)\"\" \"About 
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given:
\n" ); document.write( "(2x^2-7x-1)/(x-2)(x^2+3)
\n" ); document.write( "i) we can write
\n" ); document.write( "(2x^2-7x-1)/(x-2)(x^2+3) = A / (x-2) + (Bx+C) / (x^2+3)
\n" ); document.write( "multiply through by common denominator (x-2)(x^2+3)
\n" ); document.write( "2x^2-7x-1 = A(x^2+3) + (Bx+C)(x-2)
\n" ); document.write( "2x^2-7x-1 = Ax^2 +A3 + Bx^2 + (Cx-B2x)+ A3-2C
\n" ); document.write( "group the x terms and the constant terms
\n" ); document.write( "2x^2-7x-1 = (A+B)x^2 + (C-B2)x + (A3-2C)
\n" ); document.write( "the only 0 in the original denominator is x = 2
\n" ); document.write( "8 -14 -1 = (A+B)4 + (C-B2)2 + (A3-C2)
\n" ); document.write( "-7 = 4A+4B+2C-4B+A3-2C
\n" ); document.write( "-7 = 7A
\n" ); document.write( "A = -1
\n" ); document.write( "reconsider
\n" ); document.write( "2x^2-7x-1 = (A+B)x^2 + (C-B2)x + (A3-2C)
\n" ); document.write( "therefore
\n" ); document.write( "A+B = 2 and B=3
\n" ); document.write( "C-2B = -7
\n" ); document.write( "C = -1
\n" ); document.write( "then the decomposition is
\n" ); document.write( "(2x^2-7x-1)/(x-2)(x^2+3) = -1 / (x-2) + (3x-1) / (x^2+3)
\n" ); document.write( "ii) f(x) = -1*(x-2)^-1 + (3x-1)*(x^2+3)^-1 \n" ); document.write( "
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