document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119164>Question 119164</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm learning to divide polynomials by polynomials.  I have been able to work several problems, but then get stuck as they progress.  For example:\r<BR>\n" );
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document.write( "25x to the 5th power take away x to the third power take away 8x take away 2x square divided by 5xsquare take away 4x.  I have placed the dividend and divisor in descending order and even tried adding 0x to the fourth power in the dividend.  I seem to get the first half correct, but can't get the latter half.  What am I doing wrong?  \r<BR>\n" );
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document.write( "Jennifer </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87273 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=87273\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have done the first steps correctly, that is you put the terms in decending order of degree and put a zero coefficient placeholder in for the missing 4th degree term.  I presume you ended up with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825x%5E5%2B0x%5E4-x%5E3-2x%5E2-8x%29%2F%285x%5E2-4x%29 ALIGN=MIDDLE  ALT=\"%2825x%5E5%2B0x%5E4-x%5E3-2x%5E2-8x%29%2F%285x%5E2-4x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The next step is to factor the denominator.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2-4x=x%285x-4%29 ALIGN=MIDDLE  ALT=\"5x%5E2-4x=x%285x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now your problem looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825x%5E5%2B0x%5E4-x%5E3-2x%5E2-8x%29%2Fx%285x-4%29 ALIGN=MIDDLE  ALT=\"%2825x%5E5%2B0x%5E4-x%5E3-2x%5E2-8x%29%2Fx%285x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But notice that there is an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in every term of the numerator which means you can divide through by the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the denominator to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825x%5E4%2B0x%5E3-x%5E2-2x-8%29%2F%285x-4%29 ALIGN=MIDDLE  ALT=\"%2825x%5E4%2B0x%5E3-x%5E2-2x-8%29%2F%285x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now your polynomial long division shouldn't be so ugly.  I don't know how to render the process on this site, so I'll just talk you through it.  Write back if you have trouble understanding.\r<BR>\n" );
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document.write( "Step 1:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E4 ALIGN=MIDDLE  ALT=\"25x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3 ALIGN=MIDDLE  ALT=\"5x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3 ALIGN=MIDDLE  ALT=\"5x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term of your quotient polynomial.\r<BR>\n" );
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document.write( "Step 2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3 ALIGN=MIDDLE  ALT=\"5x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20x%5E3 ALIGN=MIDDLE  ALT=\"-20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3 ALIGN=MIDDLE  ALT=\"5x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E4 ALIGN=MIDDLE  ALT=\"25x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E4-20x%5E3 ALIGN=MIDDLE  ALT=\"25x%5E4-20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is your first partial product.\r<BR>\n" );
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document.write( "Step 3:  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E4-20x%5E3 ALIGN=MIDDLE  ALT=\"25x%5E4-20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the first two terms of the dividend polynomial giving you <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%5E4%2B20x%5E3 ALIGN=MIDDLE  ALT=\"0x%5E4%2B20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 4:  Bring down the next term, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2 ALIGN=MIDDLE  ALT=\"-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to form your next partial dividend <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x%5E3-x%5E2 ALIGN=MIDDLE  ALT=\"20x%5E3-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 6:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x%5E3 ALIGN=MIDDLE  ALT=\"20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times, so this is the next term of your quotient polynomial.  So far, your quotient should look like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3%2B4x%5E2 ALIGN=MIDDLE  ALT=\"5x%5E3%2B4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 7:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16x%5E2 ALIGN=MIDDLE  ALT=\"-16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x%5E3 ALIGN=MIDDLE  ALT=\"20x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x%5E3-16x%5E2 ALIGN=MIDDLE  ALT=\"20x%5E3-16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is your next partial product.\r<BR>\n" );
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document.write( "Step 8:  Subtract this from your previous partial dividend, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x%5E3-x%5E2 ALIGN=MIDDLE  ALT=\"20x%5E3-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (remembering to change the sign and add when you subtract), to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%5E3%2B15x%5E2 ALIGN=MIDDLE  ALT=\"0x%5E3%2B15x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 9:  Bring down the next term of the dividend, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x ALIGN=MIDDLE  ALT=\"-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to form your next partial dividend, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2-2x ALIGN=MIDDLE  ALT=\"15x%5E2-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 10:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2 ALIGN=MIDDLE  ALT=\"15x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > times giving you the next term of your quotient.  Your quotient should now look like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3%2B4x%5E2%2B3x ALIGN=MIDDLE  ALT=\"5x%5E3%2B4x%5E2%2B3x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 11:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12x ALIGN=MIDDLE  ALT=\"-12x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2 ALIGN=MIDDLE  ALT=\"15x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2-12x ALIGN=MIDDLE  ALT=\"15x%5E2-12x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is your next partial product.\r<BR>\n" );
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document.write( "Step 12:  Subtract this from your previous partial dividend, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2-2x ALIGN=MIDDLE  ALT=\"15x%5E2-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, resulting in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%5E2%2B10x ALIGN=MIDDLE  ALT=\"0x%5E2%2B10x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 13:  Bring down the last term, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to form your next partial dividend <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x-8 ALIGN=MIDDLE  ALT=\"10x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 14:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x ALIGN=MIDDLE  ALT=\"10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the last term of your quotient which should look like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3%2B4x%5E2%2B3x%2B2 ALIGN=MIDDLE  ALT=\"5x%5E3%2B4x%5E2%2B3x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 15:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x ALIGN=MIDDLE  ALT=\"10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > so your next partial product is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x-8 ALIGN=MIDDLE  ALT=\"10x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 16:  Subtract.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810x-8%29-%2810x-8%29=0 ALIGN=MIDDLE  ALT=\"%2810x-8%29-%2810x-8%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  There is no remainder, so you are done.\r<BR>\n" );
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document.write( "Hope this helps.<BR>\n" );
document.write( "John </SPAN>\n" );
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