document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=211198>Question 211198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor the following sum of 2 cubes, please  x^3+64y^3.\r<BR>\n" );
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document.write( "Factor 4r^2+21r+5\r<BR>\n" );
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document.write( "Factor if possible 9-12b+4b^2\r<BR>\n" );
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document.write( "Factor completely 3s^2-10s+8\r<BR>\n" );
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document.write( "THANK YOU!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #159565 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=159565\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first two to get you going\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B64y%5E3 ALIGN=MIDDLE  ALT=\"x%5E3%2B64y%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%5E3%2B%284y%29%5E3 ALIGN=MIDDLE  ALT=\"%28x%29%5E3%2B%284y%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%5E3 ALIGN=MIDDLE  ALT=\"%28x%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64y%5E3 ALIGN=MIDDLE  ALT=\"64y%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284y%29%5E3 ALIGN=MIDDLE  ALT=\"%284y%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4y%29%28%28x%29%5E2-%28x%29%284y%29%2B%284y%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%2B4y%29%28%28x%29%5E2-%28x%29%284y%29%2B%284y%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor by using the sum of cubes formula. Remember the <a  rel=nofollow HREF=\"http://www.purplemath.com/modules/specfact2.htm\">sum of cubes formula</a> is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29 ALIGN=MIDDLE  ALT=\"A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4y%29%28x%5E2-4xy%2B16y%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%2B4y%29%28x%5E2-4xy%2B16y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B64y%5E3 ALIGN=MIDDLE  ALT=\"x%5E3%2B64y%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4y%29%28x%5E2-4xy%2B16y%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%2B4y%29%28x%5E2-4xy%2B16y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B64y%5E3=%28x%2B4y%29%28x%5E2-4xy%2B16y%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E3%2B64y%5E3=%28x%2B4y%29%28x%5E2-4xy%2B16y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "Looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5E2%2B21r%2B5 ALIGN=MIDDLE  ALT=\"4r%5E2%2B21r%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that the first coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the second coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now multiply the first coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the last term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%285%29=20 ALIGN=MIDDLE  ALT=\"%284%29%285%29=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now the question is: what two whole numbers multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous product) <font size=4><b>and</b></font> add to the second coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous product).\r<BR>\n" );
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document.write( "Factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "1,2,4,5,10,20\r<BR>\n" );
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document.write( "-1,-2,-4,-5,-10,-20\r<BR>\n" );
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document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r<BR>\n" );
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document.write( "These factors pair up and multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "1*20 = 20<BR>\n" );
document.write( "2*10 = 20<BR>\n" );
document.write( "4*5 = 20<BR>\n" );
document.write( "(-1)*(-20) = 20<BR>\n" );
document.write( "(-2)*(-10) = 20<BR>\n" );
document.write( "(-4)*(-5) = 20\r<BR>\n" );
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document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD   align=\"center\"><font color=red>1</font></TD><TD   align=\"center\"><font color=red>20</font></TD><TD   align=\"center\"><font color=red>1+20=21</font></TD></tr><tr><TD   align=\"center\"><font color=black>2</font></TD><TD   align=\"center\"><font color=black>10</font></TD><TD   align=\"center\"><font color=black>2+10=12</font></TD></tr><tr><TD   align=\"center\"><font color=black>4</font></TD><TD   align=\"center\"><font color=black>5</font></TD><TD   align=\"center\"><font color=black>4+5=9</font></TD></tr><tr><TD   align=\"center\"><font color=black>-1</font></TD><TD   align=\"center\"><font color=black>-20</font></TD><TD   align=\"center\"><font color=black>-1+(-20)=-21</font></TD></tr><tr><TD   align=\"center\"><font color=black>-2</font></TD><TD   align=\"center\"><font color=black>-10</font></TD><TD   align=\"center\"><font color=black>-2+(-10)=-12</font></TD></tr><tr><TD   align=\"center\"><font color=black>-4</font></TD><TD   align=\"center\"><font color=black>-5</font></TD><TD   align=\"center\"><font color=black>-4+(-5)=-9</font></TD></tr>\n" );
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document.write( "From the table, we can see that the two numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the middle coefficient).\r<BR>\n" );
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document.write( "So the two numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > both multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font size=4><b>and</b></font> add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now replace the middle term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21r ALIGN=MIDDLE  ALT=\"21r\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2B20r ALIGN=MIDDLE  ALT=\"r%2B20r\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this shows us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2B20r=21r ALIGN=MIDDLE  ALT=\"r%2B20r=21r\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5E2%2Bhighlight%28r%2B20r%29%2B5 ALIGN=MIDDLE  ALT=\"4r%5E2%2Bhighlight%28r%2B20r%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace the second term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21r ALIGN=MIDDLE  ALT=\"21r\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2B20r ALIGN=MIDDLE  ALT=\"r%2B20r\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284r%5E2%2Br%29%2B%2820r%2B5%29 ALIGN=MIDDLE  ALT=\"%284r%5E2%2Br%29%2B%2820r%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the terms into two pairs.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%284r%2B1%29%2B%2820r%2B5%29 ALIGN=MIDDLE  ALT=\"r%284r%2B1%29%2B%2820r%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the first group.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%284r%2B1%29%2B5%284r%2B1%29 ALIGN=MIDDLE  ALT=\"r%284r%2B1%29%2B5%284r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B5%29%284r%2B1%29 ALIGN=MIDDLE  ALT=\"%28r%2B5%29%284r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. Or factor out the common term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%2B1 ALIGN=MIDDLE  ALT=\"4r%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5E2%2B21r%2B5 ALIGN=MIDDLE  ALT=\"4r%5E2%2B21r%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B5%29%284r%2B1%29 ALIGN=MIDDLE  ALT=\"%28r%2B5%29%284r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5E2%2B21r%2B5=%28r%2B5%29%284r%2B1%29 ALIGN=MIDDLE  ALT=\"4r%5E2%2B21r%2B5=%28r%2B5%29%284r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Note: you can check the answer by expanding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B5%29%284r%2B1%29 ALIGN=MIDDLE  ALT=\"%28r%2B5%29%284r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5E2%2B21r%2B5 ALIGN=MIDDLE  ALT=\"4r%5E2%2B21r%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or by graphing the original expression and the answer (the two graphs should be identical).<BR>\n" );
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