document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=26102>Question 26102</A>: <I><SPAN STYLE=\"word-break: break-all;\"> f(x)=4+x/(1+2x)(1-x)^2<BR>\n" );
document.write( "a) Express F(x) in partial fractions<BR>\n" );
document.write( "b) Given that |x|<0.5, expand f(x) in ascending powers of x, up to and including the term x^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #14031 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=14031\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) Express F(x) in partial fractions\r<BR>\n" );
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document.write( "f(x)=(4+x)/(1+2x)(1-x)^2<BR>\n" );
document.write( "= A/(1+2X) + B/(1-X) + C /(1-X)^2<BR>\n" );
document.write( "PUT 2X+1=0 OR X=-1/2=-0.5 TO GET A <BR>\n" );
document.write( "A=(4-0.5)/(1+0.5)^2=3.5/2.25=14/9<BR>\n" );
document.write( "AND PUT 1-X =0 OR X=1 TO GET C <BR>\n" );
document.write( "C=(4+1)/(1+2*1)=5/3....HENCE....<BR>\n" );
document.write( "f(x)=(4+x)/(1+2x)(1-x)^2 = A/(1+2X) + B/(1-X) + C /(1-X)^2<BR>\n" );
document.write( "=14/9*(1+2X) + B / (1-X) + 5 /3*(1-X)^2<BR>\n" );
document.write( "NOW PUT X=0 TO GET B<BR>\n" );
document.write( "4+0=14/9 +B +5/3....OR ....B=7/9<BR>\n" );
document.write( "HENCE WE GET <BR>\n" );
document.write( "f(x)=(4+x)/(1+2x)(1-x)^2 = A/(1+2X) + B/(1-X) + C /(1-X)^2<BR>\n" );
document.write( "=14/9*(1+2X) + 7 /9*(1-X) + 5 /3*(1-X)^2\r<BR>\n" );
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document.write( "b) Given that |x|<0.5, expand f(x) in ascending powers of x, up to and including the term x^3<BR>\n" );
document.write( "F(X)=14/9*(1+2X) + 7 /9*(1-X) + 5 /3*(1-X)^2<BR>\n" );
document.write( "=(14/9)(1+2X)^(-1)+(7/9)*(1-X)^(-1)+(5/3)*(1-X)^(-2)<BR>\n" );
document.write( "=(14/9)(1-2X+4X^2-8X^3)+(7/9)(1+X+X^2+X^3)+(5/3)(1+2X+3X^2+4X^3)<BR>\n" );
document.write( "REST YOU CAN SIMPLIFY IF NEEDED OR IT CAN BE LEFT AS SUCH..NOTE THAT THE EXPANSIONS WE DID UP TO X^3 AS DESIRED WILL HOLD GOOD ONLY IF |2X|<1...OR...|X|<0.5..HENCE THE NECESSITY FOR THE DATA THAT |X|<0.5 IN THE PROBLEM<BR>\n" );
document.write( " </SPAN>\n" );
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