document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=611509>Question 611509</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the partial fraction decomposition for each rational expression.\r<BR>\n" );
document.write( "\n" );
document.write( "(11-2x)/(x^2-8x+16) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #384983 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=384983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Finding partial fractions essentially the process of \"un-adding\" fractions.<br><BR>\n" );
document.write( "As you know, when fractions are added the denominators must be the same. And the matching denominator was the least common multiple of the denominators of the fractions to be added. So the denominator we have, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E-8x%2B16 ALIGN=MIDDLE  ALT=\"x%5E-8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the least common multiple of the fractions that we added to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811-2x%29%2F%28x%5E2-8x%2B16%29 ALIGN=MIDDLE  ALT=\"%2811-2x%29%2F%28x%5E2-8x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "So we interested in finding what denominators could there have been to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the least common multiple. For this we need to see the factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16+=+%28x-4%29%28x-4%29 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16+=+%28x-4%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With these factors the fractions we could have that would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the least common multiple would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2F%28x-4%29+%2B+B%2F%28%28x-4%29%28x-4%29%29 ALIGN=MIDDLE  ALT=\"A%2F%28x-4%29+%2B+B%2F%28%28x-4%29%28x-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we just have to figure out what \"A\" and what \"B\" would make these two fractions add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811-2x%29%2F%28x%5E2-8x%2B16%29 ALIGN=MIDDLE  ALT=\"%2811-2x%29%2F%28x%5E2-8x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To do this we first add them! Of course we need a common denominator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2F%28x-4%29%29%28%28x-4%29%2F%28x-4%29%29+%2B+B%2F%28%28x-4%29%28x-4%29%29 ALIGN=MIDDLE  ALT=\"%28A%2F%28x-4%29%29%28%28x-4%29%2F%28x-4%29%29+%2B+B%2F%28%28x-4%29%28x-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28Ax-4A%29%2F%28%28x-4%29%28x-4%29%29+%2B+B%2F%28%28x-4%29%28x-4%29%29 ALIGN=MIDDLE  ALT=\"%28Ax-4A%29%2F%28%28x-4%29%28x-4%29%29+%2B+B%2F%28%28x-4%29%28x-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we add:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28Ax-4A%2BB%29%2F%28%28x-4%29%28x-4%29%29 ALIGN=MIDDLE  ALT=\"%28Ax-4A%2BB%29%2F%28%28x-4%29%28x-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This fraction is supposed to be equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811-2x%29%2F%28x%5E2-8x%2B16%29 ALIGN=MIDDLE  ALT=\"%2811-2x%29%2F%28x%5E2-8x%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The denominators are the same so the numerators must be the same. So:<BR>\n" );
document.write( "Ax - 4A + B = 11 - 2x<br><BR>\n" );
document.write( "From this we can, believe it or not, figure out the \"A\" and the \"B\". The x term on the left is Ax and on the right it is -2x. They have to match so A = -2.<br><BR>\n" );
document.write( "The constant term on the left is -4A + B and on the right it is 11. These have to match also. So -4A + B = 11. Using the found value for A we get:<BR>\n" );
document.write( "-4(-2) + B = 11<BR>\n" );
document.write( "Solving for B...<BR>\n" );
document.write( "8 + B = 11<BR>\n" );
document.write( "B = 3<br><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811-2x%29%2F%28x%5E2-8x%2B16%29+=+%28-2%29%2F%28x-4%29+%2B+3%2F%28%28x-4%29%28x-4%29%29 ALIGN=MIDDLE  ALT=\"%2811-2x%29%2F%28x%5E2-8x%2B16%29+=+%28-2%29%2F%28x-4%29+%2B+3%2F%28%28x-4%29%28x-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );