document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=535799>Question 535799</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely,\r<BR>\n" );
document.write( "\n" );
document.write( "11x(4x-3)-6(4x-3)=\r<BR>\n" );
document.write( "\n" );
document.write( "2x(x-5)-(x-5)\r<BR>\n" );
document.write( "\n" );
document.write( "3x^3-4x^2+6x-8=\r<BR>\n" );
document.write( "\n" );
document.write( "xy+2x-y-2=\r<BR>\n" );
document.write( "\n" );
document.write( "2x^2+2x-24= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #352017 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=352017\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x%284x-3%29-6%284x-3%29 ALIGN=MIDDLE  ALT=\"11x%284x-3%29-6%284x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a common factor, present in both terms. Getting that common factor out is applying the distributive property backwards, like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x%284x-3%29-6%284x-3%29=%2811x-6%29%284x-3%29 ALIGN=MIDDLE  ALT=\"11x%284x-3%29-6%284x-3%29=%2811x-6%29%284x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%28x-5%29-%28x-5%29 ALIGN=MIDDLE  ALT=\"2x%28x-5%29-%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Same thing here<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%28x-5%29-%28x-5%29=2x%28x-5%29-1%2A%28x-5%29=%282x-1%29%28x-5%29 ALIGN=MIDDLE  ALT=\"2x%28x-5%29-%28x-5%29=2x%28x-5%29-1%2A%28x-5%29=%282x-1%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E3-4x%5E2%2B6x-8 ALIGN=MIDDLE  ALT=\"3x%5E3-4x%5E2%2B6x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > This polynomial is best factored by grouping. There are four terms. Look for a factor common to two of those terms, like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and a factor common to the other two terms. If the polynomial was an exercise in factoring, it should work. (just don't expect it to work for every polynomial.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E3-4x%5E2%2B6x-8=%283x%5E3%2B6x%29-%284x%5E2%2B8%29=3x%28x%5E2%2B2%29-4%28x%5E2%2B2%29 ALIGN=MIDDLE  ALT=\"3x%5E3-4x%5E2%2B6x-8=%283x%5E3%2B6x%29-%284x%5E2%2B8%29=3x%28x%5E2%2B2%29-4%28x%5E2%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Do you see now another common factor? Keep factoring.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%28x%5E2%2B2%29-4%28x%5E2%2B2%29=%283x-4%29%28x%5E2%2B2%29 ALIGN=MIDDLE  ALT=\"3x%28x%5E2%2B2%29-4%28x%5E2%2B2%29=%283x-4%29%28x%5E2%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that's your polynomial fully factored.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2B2x-y-2 ALIGN=MIDDLE  ALT=\"xy%2B2x-y-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is another 4-term factor-by-grouping problem. By the way, there are two ways to group. They both work (unless you make a mistake). You may see only one way, but that would be enough.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2B2x-y-2=%28xy-y%29%2B%282x-2%29=y%28x-1%29%2B2%28x-1%29=%28y%2B2%29%28x-1%29+ ALIGN=MIDDLE  ALT=\"xy%2B2x-y-2=%28xy-y%29%2B%282x-2%29=y%28x-1%29%2B2%28x-1%29=%28y%2B2%29%28x-1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B2x-24 ALIGN=MIDDLE  ALT=\"2x%5E2%2B2x-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >is a tougher factoring problem. If it can ve facored with nice integer numbers, it's going to end up being<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B2x-24=%282x%2Ba%29%28x%2Bb%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B2x-24=%282x%2Ba%29%28x%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ab=-24 ALIGN=MIDDLE  ALT=\"a%2Ab=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That gives you a bunch of choices for a and b because there are several pairs if factors whose products yield 24, and then you have a choice of which one to give the minus sign to. You know that the product term in x is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2bx%2Bax=%282b%2Ba%29x=2x ALIGN=MIDDLE  ALT=\"2bx%2Bax=%282b%2Ba%29x=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2Ba=2 ALIGN=MIDDLE  ALT=\"2b%2Ba=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I would try pairs like 3 and 8 (3x8=24) or 4 and 6 (4x6=24) rather than 2 and 12  or 1 and 24, because you need to add to 2, which is a small number.<BR>\n" );
document.write( "Trying 8 and -3: 2x8-3=13 does not work,<BR>\n" );
document.write( "but 2x(-3)+8=2 does. Lucky first guess!<BR>\n" );
document.write( "Let's verify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B8%29%28x-3%29=2x%5E2-6x%2B8x-24=2x%5E2%2B2x-24 ALIGN=MIDDLE  ALT=\"%282x%2B8%29%28x-3%29=2x%5E2-6x%2B8x-24=2x%5E2%2B2x-24\"   DISABLED_style= \"max-width:90%; height:auto;\" > We got it! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );