document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=198454>Question 198454</A>: <I><SPAN STYLE=\"word-break: break-all;\"> why do we factor polynomial equations? what are the answers called? how do t hose answers help me graph the equation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #148955 by </B><A HREF=/tutors/aboutme.mpl?userid=arallie><B>arallie(162)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=arallie><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=148955\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> why do we factor polynomial equations?\r<BR>\n" );
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document.write( "Well, factoring ISN'T important to most people in everyday life. I <BR>\n" );
document.write( "mean shopping and cleaning and cooking and going to the movies. But <BR>\n" );
document.write( "many occupations use different kinds of mathematics, ranging from <BR>\n" );
document.write( "accountants to carpenters to scientists and engineers to people who <BR>\n" );
document.write( "work to protect the environment.\r<BR>\n" );
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document.write( "Many of them will sometimes need to use factoring, but factoring isn't <BR>\n" );
document.write( "a goal in itself. Factoring is used to solve different kinds of <BR>\n" );
document.write( "problems. I think you might want to know why you should learn <BR>\n" );
document.write( "factoring if you aren't ever going to use it \"in real life.\"  \r<BR>\n" );
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document.write( "One answer to your question is that most of us don't know what we will <BR>\n" );
document.write( "be doing in real life until it happens to us. Sometimes we plan for it <BR>\n" );
document.write( "and sometimes it takes us by surprise. But it is a good idea to be <BR>\n" );
document.write( "prepared. If you don't know ANY mathematics then there are hundreds, <BR>\n" );
document.write( "maybe thousands of jobs that you won't be able to do. For most of <BR>\n" );
document.write( "these jobs mathematics isn't the main point of the job, it is just one <BR>\n" );
document.write( "of the many tools that are used. So if you don't know mathematics you <BR>\n" );
document.write( "may be losing the opportunity to do something that you would find <BR>\n" );
document.write( "exciting and worthwhile.\r<BR>\n" );
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document.write( "And now to the mathematical part of your question: how is factoring <BR>\n" );
document.write( "used?\r<BR>\n" );
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document.write( "I can think of two important uses of factoring.  \r<BR>\n" );
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document.write( "One is to make complicated things look simpler. For example,\r<BR>\n" );
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document.write( "   I don't have any idea what this fraction 1848 / 16632 means.  <BR>\n" );
document.write( "   The numbers are too large.  But if I factor the numerator and    <BR>\n" );
document.write( "   denominator I get:  (3)(8)(7)(11) / (8)(11)(7)(9) and I can <BR>\n" );
document.write( "   see that the 3, the 8, the 7, and a 5 are factors of both the  <BR>\n" );
document.write( "   numerator and denominator.  Since 3/3, 8/8, 7/7 and 11/11 are all  <BR>\n" );
document.write( "   equal to 1 the fraction reduces to 1/3, which is a lot easier to  <BR>\n" );
document.write( "   understand and to compute with.\r<BR>\n" );
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document.write( "It's the same with complex algebraic fractions. (x^4 - x^2)/(x^3+x^2) <BR>\n" );
document.write( "looks REALLY complicated, but the numerator factors into x^2(x-1)(x+1) <BR>\n" );
document.write( "and the denominator factors into x^2(x+1), and since x^2/x^2 and <BR>\n" );
document.write( "(x+1)/(x+1) are both equal to 1, this fraction simplifies to x-1 . . . <BR>\n" );
document.write( "a LOT easier to work with.\r<BR>\n" );
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document.write( "A second important use of factoring is in solving equations. You don't <BR>\n" );
document.write( "need to factor to solve 2x+3 = 5 ... linear equations use a different <BR>\n" );
document.write( "method. And you don't need to factor second degree equations because <BR>\n" );
document.write( "you can use the Quadratic Formula (although factoring is often MUCH <BR>\n" );
document.write( "easier!). But if you need to solve equations where the degree of the <BR>\n" );
document.write( "highest term is more than 2 then you really have no choice at all <BR>\n" );
document.write( "because you don't have formulas for most of them.\r<BR>\n" );
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document.write( "Here is an example of a hard equation to solve:  \r<BR>\n" );
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document.write( "   x^5 - 4x^4 - 12x^3 = 0\r<BR>\n" );
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document.write( "But if you factor it completely you get (x^3)(x-6)(x+2) = 0 and now it <BR>\n" );
document.write( "is easy, because this says that a product of things turns out to be <BR>\n" );
document.write( "equal to zero. If you multiply, the only way to get zero as an answer <BR>\n" );
document.write( "would be if you multiplied by zero. So one of the three factors has <BR>\n" );
document.write( "to be zero.\r<BR>\n" );
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document.write( "    If x^3 = 0 then x = 0<BR>\n" );
document.write( "    If x-6 = 0 then x = 6<BR>\n" );
document.write( "    If x+2 = 0 then x = -2\r<BR>\n" );
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document.write( "So the solutions to this equation are x = 0 or 6 or -2.\r<BR>\n" );
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document.write( "I hope that helps.<BR>\n" );
document.write( "--Doctor Sam (not me)\r<BR>\n" );
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document.write( "what are the answers called?<BR>\n" );
document.write( "Simply, factors\r<BR>\n" );
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document.write( "how do those answers help me graph the equation?<BR>\n" );
document.write( "I you mark the factors at (x,0), always y=0, and graph the mid-point, which is ((x[1]+x[2])/2,c) where x[1] and x[2] are the factors and c is the part of the simplified equation that has no variable, you will then begin to see a graph forming. </SPAN>\n" );
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