document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=239120>Question 239120</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the LCM <BR>\n" );
document.write( "z^3+4z^2+4z, z^2-4z<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #175634 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=175634\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Whether you are working with just plain numbers or with expressions, you find LCM's the same way: You factor the numbers or expressions and use those factors to find the LCM.<br><BR>\n" );
document.write( "So we start by factoring and you always start factoring with the Greatest Common Factor (GCF). The GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2B4z%5E2%2B4z ALIGN=MIDDLE  ALT=\"z%5E3%2B4z%5E2%2B4z\"   DISABLED_style= \"max-width:90%; height:auto;\" > is z so we can factor it into:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2B4z%5E2%2B4z+=+z%28z%5E2%2B4z%2B4%29 ALIGN=MIDDLE  ALT=\"z%5E3%2B4z%5E2%2B4z+=+z%28z%5E2%2B4z%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then the second factor fits the perfect square trinomial pattern, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%2B+2ab+%2B+b%5E2+=+%28a%2Bb%29%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+%2B+2ab+%2B+b%5E2+=+%28a%2Bb%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where the \"a\" is \"z\" and the \"b\" is \"2\". So it will factor into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"%28z%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2B4z%5E2%2B4z+=+z%28z%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"z%5E3%2B4z%5E2%2B4z+=+z%28z%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we'll factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-4z ALIGN=MIDDLE  ALT=\"z%5E2-4z\"   DISABLED_style= \"max-width:90%; height:auto;\" >. (Was this supposed to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-4z ALIGN=MIDDLE  ALT=\"z%5E3-4z\"   DISABLED_style= \"max-width:90%; height:auto;\" >? If so, the problem is a bit more interesting. Just in case, I'll solve using this, too, at the end.) Again we start with the GCF (which is z again):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-4z+=+z%28z-4%29 ALIGN=MIDDLE  ALT=\"z%5E2-4z+=+z%28z-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And this is all the factoring we can do.<br>So we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2B4z%5E2%2B4z+=+z%28z%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"z%5E3%2B4z%5E2%2B4z+=+z%28z%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-4z+=+z%28z-4%29 ALIGN=MIDDLE  ALT=\"z%5E2-4z+=+z%28z-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find the LCM you find the product of <i>every</i> different factor. In the case of any common factors, use the higher exponent on those factors. Between these two factored expressions, we have 3 different factors: z, z+2 and z-4. z is a factor common to both. In both factored expressions, the exponent on z is 1 so the \"higher\" exponent is 1. So the LCM is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28z%2B2%29%5E2%28z-4%29 ALIGN=MIDDLE  ALT=\"z%28z%2B2%29%5E2%28z-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If your answer is supposed to be simplified, you'll need to multiply this out.<br><BR>\n" );
document.write( "In case the second expression was supposed to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-4z ALIGN=MIDDLE  ALT=\"z%5E3-4z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then it factors this way:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-4z+=+z%28z%5E2+-+4%29 ALIGN=MIDDLE  ALT=\"z%5E3-4z+=+z%28z%5E2+-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The second factor is a difference of squares so it factors:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-4z+=+z%28z%2B2%29%28z-2%29 ALIGN=MIDDLE  ALT=\"z%5E3-4z+=+z%28z%2B2%29%28z-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the LCM between <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2B4z%5E2%2B4z+=+z%28z%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"z%5E3%2B4z%5E2%2B4z+=+z%28z%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-4z+=+z%28z%2B2%29%28z-2%29 ALIGN=MIDDLE  ALT=\"z%5E3-4z+=+z%28z%2B2%29%28z-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%28z%2B2%29%5E2%28z-2%29 ALIGN=MIDDLE  ALT=\"z%28z%2B2%29%5E2%28z-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Note how, with the z+2 factor, we used the higher exponent in the LCM). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );