document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185961>Question 1185961</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Expand (1/2-2x)^5 up to the term in x^3. If the coefficient of x^2 in the expansion of (1+ax+3x^2)(1/2-2x)^5 is 13/2, find the coefficient of 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 #816828 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=816828\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "(1) Expand (1/2-2x)^5 up to the term in x^3<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5E5+%2B+%285%29%28%281%2F2%29%5E4%29%28-2x%29%5E1+%2B+%2810%29%28%281%2F2%29%5E3%29%28-2x%29%5E2+%2B+10%28%281%2F2%29%5E2%29%28-2x%29%5E3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > + ...<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F32%29+-+%285%2F8%29x+%2B+5x%5E2+-+20x%5E3 ALIGN=MIDDLE  ALT=\"%281%2F32%29+-+%285%2F8%29x+%2B+5x%5E2+-+20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > + ...<br><BR>\n" );
document.write( "(2) Find a, given that the coefficient of x^2 in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bax%2B3x%5E2%29%281%2F2-2x%29%5E5 ALIGN=MIDDLE  ALT=\"%281%2Bax%2B3x%5E2%29%281%2F2-2x%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 13/2<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bax%2B3x%5E2%29%28%281%2F32%29+-+%285%2F8%29x+%2B+5x%5E2+-+20x%5E3%29%29 ALIGN=MIDDLE  ALT=\"%281%2Bax%2B3x%5E2%29%28%281%2F32%29+-+%285%2F8%29x+%2B+5x%5E2+-+20x%5E3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The contributions to the x^2 coefficient come from the constant term in the first polynomial times the coefficient of x^2 in the second, the coefficient of the x term in the first polynomial times the coefficient of the x term in the second, and the coefficient of the x^2 term in the first polynomial times the constant term in the second:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F32%29%283%29%2B%28-5%2F8%29%28a%29%2B%285%29%281%29+=+13%2F2 ALIGN=MIDDLE  ALT=\"%281%2F32%29%283%29%2B%28-5%2F8%29%28a%29%2B%285%29%281%29+=+13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F32-%285%2F8%29a%2B5+=+13%2F2 ALIGN=MIDDLE  ALT=\"3%2F32-%285%2F8%29a%2B5+=+13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-20a%2B160+=+208 ALIGN=MIDDLE  ALT=\"3-20a%2B160+=+208\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20a=-45 ALIGN=MIDDLE  ALT=\"20a=-45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-9%2F4 ALIGN=MIDDLE  ALT=\"a=-9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: a=-9/4<br><BR>\n" );
document.write( "(3) Find the coefficient of x^3 in (1+ax+3x^2)(1/2-2x)^5<br><BR>\n" );
document.write( "The contributions to the x^3 coefficient come from the constant term in the first polynomial times the coefficient of the x^3 term in the second, the coefficient of the x term in the first polynomial times the coefficient of the x^2 term in the second, and the coefficient of the x^2 term in the first polynomial times the coefficient of the x term in the second:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-20%29%281%29%2B%285%29%28-9%2F4%29%2B%28-5%2F8%29%283%29+=+%28-160-90-15%29%2F8+=+-265%2F8 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: -265/8<br><BR>\n" );
document.write( "The answers are confirmed with this completely expanded polynomial as calculated on wolframalpha.com:<br><BR>\n" );
document.write( "1/32 - (89 x)/128 + (13 x^2)/2 - (265 x^3)/8 + 100 x^4 - 182 x^5 + 192 x^6 - 96 x^7<br><BR>\n" );
document.write( " </SPAN>\n" );
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