document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208433>Question 1208433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the LCM of the given polynomials.\r<BR>\n" );
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document.write( "x^2 - x- 12; x^2 - 8x + 16\r<BR>\n" );
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document.write( "Let me see.\r<BR>\n" );
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document.write( "x^2 - x - 12 = (x - 4)(x + 3)\r<BR>\n" );
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document.write( "x^2 - 8x + 16 = (x - 4)(x - 4) = (x - 4)^2\r<BR>\n" );
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document.write( "LCM = (x - 4)(x + 3)\r<BR>\n" );
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document.write( "You say?\r<BR>\n" );
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document.write( "Question:\r<BR>\n" );
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document.write( "What about if there are 3 given polynomials?\r<BR>\n" );
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document.write( "Sample:\r<BR>\n" );
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document.write( "Given x^2 + 4x + 4; x^3 + 2x^2; (x + 2)^3, find the LCM. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846855 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=846855\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's look at an example involving 3 numbers.<BR>\n" );
document.write( "Question: Find the LCM of 12,18, and 80\r<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "Find the prime factorization of each<BR>\n" );
document.write( "12 = 2*2*3<BR>\n" );
document.write( "18 = 2*3*3<BR>\n" );
document.write( "80 = 2*2*2*2*5\r<BR>\n" );
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document.write( "The unique primes are 2,3,5<BR>\n" );
document.write( "2 shows up at most 4 times, which means 2^4 is a factor of the LCM.<BR>\n" );
document.write( "3 shows up at most twice, so 3^2 is also a factor of the LCM. <BR>\n" );
document.write( "5 shows up at most once, so 5^1 is also a factor of the LCM. <BR>\n" );
document.write( "The LCM is therefore 2^4*3^2*5^1 = 16*9*5 = 720<BR>\n" );
document.write( "Various online LCM calculators can be used to verify. \r<BR>\n" );
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document.write( "With that example in mind, we can extend to polynomials.<BR>\n" );
document.write( "x^2-x-12 = (x-4)(x+3)<BR>\n" );
document.write( "x^2-8x+16 = (x-4)(x-4)\r<BR>\n" );
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document.write( "The unique prime polynomials are x-4 and x+3<BR>\n" );
document.write( "(x-4) shows up at most twice while (x+3) shows up at most once<BR>\n" );
document.write( "LCM = (x-4)^2*(x+3)<BR>\n" );
document.write( "The order of the factor doesn't matter.<BR>\n" );
document.write( "WolframAlpha is one of many tools that you can use to verify.\r<BR>\n" );
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document.write( "Now onto 3 polynomials<BR>\n" );
document.write( "x^2+4x+4 = (x+2)(x+2)<BR>\n" );
document.write( "x^3+2x^2 = x^2(x+2)<BR>\n" );
document.write( "(x+2)^3 = (x+2)(x+2)(x+2)\r<BR>\n" );
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document.write( "Unique prime polynomials: x, (x+2)<BR>\n" );
document.write( "x shows up at most twice, so x^2 is part of the LCM.<BR>\n" );
document.write( "(x+2) shows up at most three times, so (x+2)^3 is also part of the LCM.<BR>\n" );
document.write( "The LCM is x^2(x+2)^3<BR>\n" );
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