document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=704297>Question 704297</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how can i understand foiling and factoring better? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #434031 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=434031\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Factoring takes practice and it is hard work.<BR>\n" );
document.write( "The good news is that factoring gets easier with practice, and it is a very efficient way to solve some quadratic equations.<BR>\n" );
document.write( "The bad news is that factoring keeps coming up in math, from rational functions and quadratic equations all the way to calculus.<BR>\n" );
document.write( "I will show you my strategies, along with the rationale behind it.<BR>\n" );
document.write( "Understanding the rationale helps me remember procedures without having to memorize \"recipes\" that make no sense to me.<BR>\n" );
document.write( "There will be too many words, but I'll use more pictures as it gets too complicated for just words..<BR>\n" );
document.write( " <BR>\n" );
document.write( "I assume you know that the first thing to do when factoring a complicated expression is to look for common factors.<BR>\n" );
document.write( "If you have a polynomial like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-5x%5E2-6x ALIGN=MIDDLE  ALT=\"x%5E3-5x%5E2-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <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 all the terms,<BR>\n" );
document.write( "you would first \"take out the common factor\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-5x%5E2-6x=x%28x%5E2-5x-6%29 ALIGN=MIDDLE  ALT=\"x%5E3-5x%5E2-6x=x%28x%5E2-5x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "FOIL is an acronym teachers use to help you remember all the terms when multiplying two binomials.<BR>\n" );
document.write( "When multiplying the binomials <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2BS ALIGN=MIDDLE  ALT=\"F%2BS\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%2Bs ALIGN=MIDDLE  ALT=\"f%2Bs\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you get a sum of all four possible products:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28F%2BS%29%28f%2Bs%29=F%2Af%2BF%2As%2BS%2Af%2Bs%2Af ALIGN=MIDDLE  ALT=\"%28F%2BS%29%28f%2Bs%29=F%2Af%2BF%2As%2BS%2Af%2Bs%2Af\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the acronym FOIL,<BR>\n" );
document.write( "F stand for First terms multiplied make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ff ALIGN=MIDDLE  ALT=\"Ff\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "O stands for Outside terms multiplied make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Fs ALIGN=MIDDLE  ALT=\"Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I L stands for stands for Inside terms multiplied make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Sf ALIGN=MIDDLE  ALT=\"Sf\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Last term multiplied make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sf ALIGN=MIDDLE  ALT=\"sf\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Thinking of FOIL makes sure you include all four products, just once each, and nothing else.<BR>\n" );
document.write( "In <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B7%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x%2B7%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   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;\" > are the Outside terms.<BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and 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;\" > that appear together in the middle are the Inside terms<BR>\n" );
document.write( "<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;\" > is the First terms multiplied,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x ALIGN=MIDDLE  ALT=\"-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the Outside terms multiplied,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x ALIGN=MIDDLE  ALT=\"7x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the Inside terms multiplied<BR>\n" );
document.write( "After the \"foiling\" often comes the \"collecting like terms\", so after<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B7%29%28x-2%29=x%5E2-2x%2B7x-14 ALIGN=MIDDLE  ALT=\"%28x%2B7%29%28x-2%29=x%5E2-2x%2B7x-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you add together the terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x ALIGN=MIDDLE  ALT=\"-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x ALIGN=MIDDLE  ALT=\"7x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B7x=5x ALIGN=MIDDLE  ALT=\"-2x%2B7x=5x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then you have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B7%29%28x-2%29=x%5E2-2x%2B7x-14=x%5E2%2B5x-14 ALIGN=MIDDLE  ALT=\"%28x%2B7%29%28x-2%29=x%5E2-2x%2B7x-14=x%5E2%2B5x-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Factoring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5x-6 ALIGN=MIDDLE  ALT=\"x%5E2-5x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is \"unfoiling\".<BR>\n" );
document.write( "You know that the <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;\" > is the product of two first terms that must have been <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 both binomials.<BR>\n" );
document.write( "You know that the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > term is the product of two last terms, and it is not that clear what those two last terms were.<BR>\n" );
document.write( "There are several possibilities (four to be specific):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%29%28-1%29=-6 ALIGN=MIDDLE  ALT=\"%286%29%28-1%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%281%29=-6 ALIGN=MIDDLE  ALT=\"%28-6%29%281%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%28-2%29=-6 ALIGN=MIDDLE  ALT=\"%283%29%28-2%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%282%29=-6 ALIGN=MIDDLE  ALT=\"%28-3%29%282%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the foiling, the last terms of the binomial were also used as factors in the Outside and Inside products to make two terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In those two terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > the last terms of the binomial appeared as coefficients in front of x.<BR>\n" );
document.write( "The problem is that the two terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the foiling were already \"collected\" together into the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5x ALIGN=MIDDLE  ALT=\"-5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > term,<BR>\n" );
document.write( "so the coefficients were added together to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We go back to the four possibilities and see what pair adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%29%2B%28-1%29=5 ALIGN=MIDDLE  ALT=\"%286%29%2B%28-1%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%2B%281%29=-5 ALIGN=MIDDLE  ALT=\"%28-6%29%2B%281%29=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%2B%28-2%29=1 ALIGN=MIDDLE  ALT=\"%283%29%2B%28-2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%2B%282%29=-1 ALIGN=MIDDLE  ALT=\"%28-3%29%2B%282%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so the two last terms must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29 ALIGN=MIDDLE  ALT=\"%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29 ALIGN=MIDDLE  ALT=\"%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So the factoring is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5x-6=%28x-6%29%28x%2B1%29 ALIGN=MIDDLE  ALT=\"x%5E2-5x-6=%28x-6%29%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "A PICTURE:<BR>\n" );
document.write( "In the picture below, the large square represents the product <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28c%2Bd%29=ac%2Bad%2Bbc%2Bbd ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28c%2Bd%29=ac%2Bad%2Bbc%2Bbd\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It is what happens when a patio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > tiles long by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > tiles wide<BR>\n" );
document.write( "is enlarged by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > rows of tiles to the length and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > rows to the wdith.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3%2C11%2C-1%2C13%2C%0D%0Arectangle%280%2C0%2C10%2C10%29%2C%0D%0Aline%287%2C0%2C7%2C10%29%2Cline%280%2C6%2C10%2C6%29%2C%0D%0Aline%280%2C11%2C10%2C11%29%2Cline%28-1%2C0%2C-1%2C10%29%2C%0D%0Aline%280%2C10.5%2C0%2C12%29%2Cline%287%2C10.5%2C7%2C11%29%2Cline%2810%2C10.5%2C10%2C12%29%2C%0D%0Aline%28-2%2C0%2C-0.5%2C0%29%2Cline%28-1%2C6%2C-0.5%2C6%29%2Cline%28-2%2C10%2C-0.5%2C10%29%2C%0D%0Alocate%283.3%2C11%2Ca%29%2Clocate%288.3%2C11%2Cb%29%2Clocate%284.7%2C12%2C%28a%2Bb%29%29%2C%0D%0Alocate%28-0.9%2C8.5%2Cc%29%2Clocate%28-0.9%2C3.5%2Cd%29%2Clocate%28-3%2C5.5%2C%28c%2Bd%29%29%2C%0D%0Alocate%283.3%2C8.5%2Cac%29%2Clocate%288.3%2C8.5%2Cbc%29%2C%0D%0Alocate%283.3%2C3.5%2Cad%29%2Clocate%288.3%2C3.5%2Cbd%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Foiling is multiplying the side measures to get the area of the rectangle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The figure below illustrates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B2%29%285x-4%29=15x%5E2-12x%2B10x-8=15x%5E2-2x-8 ALIGN=MIDDLE  ALT=\"%283x%2B2%29%285x-4%29=15x%5E2-12x%2B10x-8=15x%5E2-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C10.5%2C-0.5%2C11.5%2C%0D%0Arectangle%280%2C0%2C10%2C10%29%2C%0D%0Aline%285%2C0%2C5%2C10%29%2Cline%280%2C5%2C10%2C5%29%2C%0D%0Alocate%282.4%2C10.7%2C3x%29%2Clocate%287.4%2C10.7%2C2%29%2Clocate%284.9%2C10.7%2C%22%2B%22%29%2C%0D%0Alocate%28-0.8%2C7.8%2C5x%29%2Clocate%28-1%2C2.8%2C-4%29%2C%0D%0Alocate%282%2C8.2%2C15x%5E2%29%2Clocate%287%2C7.8%2C10x%29%2C%0D%0Alocate%281.9%2C2.8%2C-12x%29%2Clocate%287%2C2.8%2C-8%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Factoring is figuring out the rectangle sides from the area.<BR>\n" );
document.write( "The figures below illustrates factoring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2-2x-8 ALIGN=MIDDLE  ALT=\"15x%5E2-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I will pretend that I do not know what the factors were as I try to undo the multiplication.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C10.5%2C-0.5%2C11.5%2C%0D%0Arectangle%280%2C0%2C10%2C10%29%2C%0D%0Aline%285%2C0%2C5%2C10%29%2Cline%280%2C5%2C10%2C5%29%2C%0D%0Alocate%282.4%2C10.7%2Cax%29%2Clocate%287.4%2C10.7%2Cb%29%2Clocate%284.9%2C10.7%2C%22%2B%22%29%2C%0D%0Alocate%28-0.5%2C5.3%2C%22%2B%22%29%2Clocate%28-1.2%2C7.8%2Ccx%29%2Clocate%28-0.5%2C2.8%2Cd%29%2C%0D%0Alocate%281%2C8.2%2Cacx%5E2=15x%5E2%29%2Clocate%287%2C7.8%2Cbcx%29%2C%0D%0Alocate%281.9%2C2.8%2Cadx%29%2Clocate%286.5%2C2.8%2Cbd=-8%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > If you multiply together opposite corners, you find that they are the same.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28acx%5E2%29%28bd%29=abcdx%5E2 ALIGN=MIDDLE  ALT=\"%28acx%5E2%29%28bd%29=abcdx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28bcx%29%28adx%29=abcdx%5E2 ALIGN=MIDDLE  ALT=\"%28bcx%29%28adx%29=abcdx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bcx ALIGN=MIDDLE  ALT=\"bcx\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=adx ALIGN=MIDDLE  ALT=\"adx\"   DISABLED_style= \"max-width:90%; height:auto;\" > that we are looking for must be factors of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815x%5E2%29%28-8%29=-120x%5E2 ALIGN=MIDDLE  ALT=\"%2815x%5E2%29%28-8%29=-120x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We look for pairs of factors that multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2A8=120 ALIGN=MIDDLE  ALT=\"15%2A8=120\"   DISABLED_style= \"max-width:90%; height:auto;\" > without worrying much about signs, or about 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;\" >.<BR>\n" );
document.write( "I know that 1,2,3,4,5,6,8,10,and 12 are factors because they divide 120 evenly).<BR>\n" );
document.write( "Dividing 120 by each of the small,easy to find factors, I find the matching larger factors.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120=1%2A120=2%2A60=3%2A40=4%2A30=5%2A24=6%2B20=8%2A15=10%2A12 ALIGN=MIDDLE  ALT=\"120=1%2A120=2%2A60=3%2A40=4%2A30=5%2A24=6%2B20=8%2A15=10%2A12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When I get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > I realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A10 ALIGN=MIDDLE  ALT=\"12%2A10\"   DISABLED_style= \"max-width:90%; height:auto;\" > involves the same pair of factors I had already found as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2A12 ALIGN=MIDDLE  ALT=\"10%2A12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so I know I have found all the factors.<BR>\n" );
document.write( "Because the product has a minus sign, I know that one of the factors must have a minus sign.<BR>\n" );
document.write( "Because they must add to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x ALIGN=MIDDLE  ALT=\"-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I figure that I am looking for a pair of factors that differ by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and I am going to give the minus sign to the larger factor.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-10=2 ALIGN=MIDDLE  ALT=\"12-10=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the answer<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-8=7 ALIGN=MIDDLE  ALT=\"15-8=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not work, and neither do the other pairs.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the coefficients I am looking for.<BR>\n" );
document.write( "I fill the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bcx ALIGN=MIDDLE  ALT=\"bcx\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=adx ALIGN=MIDDLE  ALT=\"adx\"   DISABLED_style= \"max-width:90%; height:auto;\" > squares with <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=10x ALIGN=MIDDLE  ALT=\"10x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It does not matter where I put each one.  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C10.5%2C-0.5%2C11.5%2C%0D%0Arectangle%280%2C0%2C10%2C10%29%2C%0D%0Aline%285%2C0%2C5%2C10%29%2Cline%280%2C5%2C10%2C5%29%2C%0D%0Alocate%282.4%2C10.7%2C%22%3F%22%29%2Clocate%287.4%2C10.7%2C%22%3F%22%29%2Clocate%284.9%2C10.7%2C%22%2B%22%29%2C%0D%0Alocate%28-0.5%2C5.3%2C%22%2B%22%29%2Clocate%28-0.5%2C7.8%2C%22%3F%22%29%2Clocate%28-0.5%2C2.8%2C%22%3F%22%29%2C%0D%0Alocate%282%2C8.2%2C15x%5E2%29%2Clocate%287%2C7.8%2C-12x%29%2C%0D%0Alocate%281.9%2C2.8%2C10x%29%2Clocate%287%2C2.8%2C-8%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I need to figure common factors for each row and column. Those common factors will replace the question marks.<BR>\n" );
document.write( "I start with what seems more obvious to me, whatever I can figure out faster.<BR>\n" );
document.write( "The <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x ALIGN=MIDDLE  ALT=\"10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the left column share <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a common factor, so I put it above the left column.<BR>\n" );
document.write( "The <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12x ALIGN=MIDDLE  ALT=\"-12x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the top row share <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a common factor, so I put it to the left of the top row. I could also have figured it, from what I already knew, as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%5E2%2F5x=3x ALIGN=MIDDLE  ALT=\"15x%5E2%2F5x=3x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The other question marks, I can figure out from what I already know:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12x%2F3x=-4 ALIGN=MIDDLE  ALT=\"-12x%2F3x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes above <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<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2F5x=2 ALIGN=MIDDLE  ALT=\"10x%2F5x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes above <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x ALIGN=MIDDLE  ALT=\"10x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C10.5%2C-0.5%2C11.5%2C%0D%0Arectangle%280%2C0%2C10%2C10%29%2C%0D%0Aline%285%2C0%2C5%2C10%29%2Cline%280%2C5%2C10%2C5%29%2C%0D%0Alocate%282.4%2C10.7%2C5x%29%2Clocate%287.4%2C10.7%2C-4%29%2Clocate%284.9%2C10.7%2C%22%2B%22%29%2C%0D%0Alocate%28-0.5%2C5.3%2C%22%2B%22%29%2Clocate%28-0.8%2C7.8%2C3x%29%2Clocate%28-1%2C2.8%2C2%29%2C%0D%0Alocate%282%2C8.2%2C15x%5E2%29%2Clocate%287%2C7.8%2C-12x%29%2C%0D%0Alocate%281.9%2C2.8%2C10x%29%2Clocate%287%2C2.8%2C-8%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "SPECIAL PRODUCTS:<BR>\n" );
document.write( "There are special products that you end up remembering after a while, and are represented by the \"formulas\" below.<BR>\n" );
document.write( "Square of a binomial:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Difference of squares:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-b%5E2=%28a%2Bb%29%28a-b%29 ALIGN=MIDDLE  ALT=\"a%5E2-b%5E2=%28a%2Bb%29%28a-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That one is easy to explain because when you FOIL, the O and I products cancel out, so your FOIL turns into FL (I cal that Florida).<BR>\n" );
document.write( "Difference of cubes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3-b%5E3=%28a%5E2%2Bab%2Bb%5E2%29%28a-b%29 ALIGN=MIDDLE  ALT=\"a%5E3-b%5E3=%28a%5E2%2Bab%2Bb%5E2%29%28a-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Sum of cubes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2Bb%5E3=%28a%5E2-ab%2Bb%5E2%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"a%5E3%2Bb%5E3=%28a%5E2-ab%2Bb%5E2%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );