document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=188497>Question 188497</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having a hard time trying to factor this equation. (x²-x+1) <BR>\n" );
document.write( "Im starting to think it is prime but I'd like to make sure. <BR>\n" );
document.write( "This is how i worked it out.<BR>\n" );
document.write( "i know that each set of parantheses will start out with \"x\". then i would figure out what times what will give you a positive one and subtract to give you a negative one. If you could please explain to me how to come about getting this answer, that would be greatly apprciated. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #141325 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=141325\">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( "The short answer is that this trinomial is prime, meaning that it is not factorable over the integers.  In fact, this one isn't factorable over the real numbers.\r<BR>\n" );
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document.write( "This might be getting a little ahead of you, given the way you phrased the question, but it is something that you will be learning in a week or two anyway.  What I'm talking about is a little trick to determine the character of the roots of a quadratic equation of the form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ax^2 + bx + c = 0\">\r<BR>\n" );
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document.write( "Take the coefficient on the <i>x</i> term and square it:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b^2\">\r<BR>\n" );
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document.write( "Multiply 4 times the lead coefficient times the constant term:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4ac\">\r<BR>\n" );
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document.write( "Find the difference of these two values: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b^2 - 4ac\">\r<BR>\n" );
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document.write( "If the result is zero, then you have a perfect square trinomial, and this would be factorable over the integers..\r<BR>\n" );
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document.write( "If the result is positive and a perfect square, then the trinomial is factorable over the integers.\r<BR>\n" );
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document.write( "If the result is positive and not a perfect square, the factors are real but irrational numbers, i.e. they involve the square root of a quantity that is not a perfect square. For the time being, you would consider such a trinomial as prime.\r<BR>\n" );
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document.write( "If the result is less than zero, then the factors are complex numbers of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha \pm \beta i\"> where <i>i</i> is the imaginary number defined by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i^2 = -1\">.  You would consider this situation prime as well.\r<BR>\n" );
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document.write( "So, just to illustrate with your problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a = 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ b = -1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c = 1\">\r<BR>\n" );
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document.write( "so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (-1)^2 - 4(1)(1) = -3 < 0\">\r<BR>\n" );
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document.write( "and your factors would contain a conjugate pair of complex numbers; they would look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - (\alpha + \beta i))(x - (\alpha - \beta i))\">\r<BR>\n" );
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document.write( "Like I said, the trinomial is prime as far as you are concerned.\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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