document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=310731>Question 310731</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When factoring a trinomial, the last digit, and middle coefficient are important.  Explain why. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #222230 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=222230\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "If by \"last digit\" you mean the constant coefficient, then yes, it is every bit as important as the first degree or linear term coefficient, which I presume you are referring to when you say \"middle coefficient.\"  However, neither is any more or less important than the lead coefficient, which is to say the coefficient on the high order term.\r<BR>\n" );
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document.write( "Also, the term 'last digit' doesn't really describe what you mean.  Consider: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ +\ 10x\ +\ 21\ =\ 0\">.  Here, the 'last digit' could be construed to mean \"1\" (and quite properly, considering the meaning of the word 'digit').  And, of course, knowing just the last digit in this case is decidedly unhelpful when attempting to factor the expression given.\r<BR>\n" );
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document.write( "\"Middle coefficient\" is rather poor terminology as well.  Who says that the first degree term will always be presented \"in the middle\"?  \r<BR>\n" );
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document.write( "Proof:\r<BR>\n" );
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document.write( "The general solution of a quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2\ +\ bx\ +\ c\ =\ 0\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \frac{-b\ \pm\ \sqrt{b^2\ -\ 4ac}}{2a}\">\r<BR>\n" );
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document.write( "Notice that all three coefficients, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> are required by the formula.  Hence, whether the quadratic trinomial is factorable over the rational numbers or not, you still must consider all three coefficients.\r<BR>\n" );
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document.write( "Even changing your question so that it reads \"When factoring a quadratic trinomial with a lead coefficient of 1,...\" doesn't change the fact that you are considering the lead coefficient simply by virtue of the fact that you have restricted it to a value of 1.\"\r<BR>\n" );
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document.write( "No disrespect to your teacher meant, but this is a very poorly worded question.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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