document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208159>Question 1208159</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that x^2 + 4 is prime.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846370 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52797)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Show that x^2 + 4 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28a%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > prime.<BR>\n" );
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document.write( "The complete formulation of the problem is\r<BR>\n" );
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document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Show that &nbsp;x^2 + 4 &nbsp;is a prime polynomial\r<BR>\n" );
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document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;over the set of polynomials with real coefficients.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I will give two proofs.\r<BR>\n" );
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document.write( "        First proof is very similar to that which I gave to other (similar) problem under this link\r\n" );
document.write( "        <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Expressions-with-variables/Expressions-with-variables.faq.question.1208160.html>https://www.algebra.com/algebra/homework/Expressions-with-variables/Expressions-with-variables.faq.question.1208160.html</A> \r\n" );
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document.write( "If x^2 + 4 is not a prime over the set of polynomials with real coefficients, \r\n" );
document.write( "then it is the product of two linear binomials\r\n" );
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document.write( "    x^2 + 4 = (ax+b)*(cx+d),\r\n" );
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document.write( "where a, b, c and d are real numbers.  Since ab must be equal to 1, the coefficients \"a\" and \"b\"\r\n" );
document.write( "must be non-zero real numbers. But then the polynomial x^2 + 4 has two real roots\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D ALIGN=MIDDLE  ALT=\"x%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-d%2Fc ALIGN=MIDDLE  ALT=\"-d%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "But, from the other hand, the discriminant of this quadratic polynomial is\r\n" );
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document.write( "    \"b^2 - 4ac\" = 0^2 - 4*1*4 = 0 - 16 = -16,\r\n" );
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document.write( "which is negative number.\r\n" );
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document.write( "A quadratic polynomial with negative discriminant can not have real roots.\r\n" );
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document.write( "This contradiction proves that given polynomial x^2 + 4 is a prime in the set of\r\n" );
document.write( "polynomial with real coefficients.\r\n" );
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document.write( "At this point, the first proof is completed.\r<BR>\n" );
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document.write( "      Second proof is more simple.\r\n" );
document.write( "      It does not use the conception of the discriminant.\r\n" );
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document.write( "If x^2 + 4 is not a prime over the set of polynomials with real coefficients, \r\n" );
document.write( "then it is the product of two linear binomials\r\n" );
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document.write( "    x^2 + 4 = (ax+b)*(cx+d),\r\n" );
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document.write( "where a, b, c and d are real numbers.  Since ab must be equal to 1, the coefficients \"a\" and \"b\"\r\n" );
document.write( "must be non-zero real numbers. But then the polynomial x^2 + 4 has two real roots\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D ALIGN=MIDDLE  ALT=\"x%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-d%2Fc ALIGN=MIDDLE  ALT=\"-d%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "From the other hand, it is easy to see that the quadratic function y = x^2 + 4\r\n" );
document.write( "is always positive and, therefore, has no real roots.\r\n" );
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document.write( "This contradiction proves the statement and completes the solution.\r\n" );
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document.write( "Solved/proved in full in two ways, for your better understanding.\r<BR>\n" );
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document.write( "<U>Comment from student</U>: Well, easy for you. Difficult for most people.\r<BR>\n" );
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document.write( "<U>My response</U>: &nbsp;&nbsp;(1) &nbsp;&nbsp;Who authorized you to speak on behalf of the most people?\r<BR>\n" );
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document.write( "You yourself authorized yourself ? - Please get the lowest possible score from me for your comment.\r<BR>\n" );
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document.write( "(2) &nbsp;&nbsp;My goal is not to make it easy for all.\r<BR>\n" );
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document.write( "My goal is to make it educative for those who will read my post and is able to make <BR>\n" );
document.write( "minimal necessary mental efforts to understand my writing.\r<BR>\n" );
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document.write( "(3) &nbsp;&nbsp;And another question: &nbsp;for what reason do you come to this forum, <BR>\n" );
document.write( "if you do not want or are not able to make even minimal microscopic mental effort <BR>\n" );
document.write( "to absorb what was just explained to you?\r<BR>\n" );
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document.write( "The idea that Mathematics should be understandable for everyone is simply crazy.<BR>\n" );
document.write( "Just like the idea that Mathematics should be taught equally to everyone.\r<BR>\n" );
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document.write( "These ideas are of the same category as teaching everyone and everybody to play <BR>\n" );
document.write( "violin or to play piano to a master's level.\r<BR>\n" );
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document.write( "If you follow these ideas, you will have revolutions and civil wars in the country <BR>\n" );
document.write( "and in society every 20-30 years.\r<BR>\n" );
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document.write( "Mathematics at a good level should only be taught to those who have an inclination <BR>\n" );
document.write( "for it, who feel the harmony of mathematics and who are ready to make the necessary <BR>\n" );
document.write( "efforts to comprehend and master it.\r<BR>\n" );
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document.write( "The same as with music, with the piano and the violin.\r<BR>\n" );
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document.write( "For everyone else - the multiplication table, the square roots and a few of logarithms.\r<BR>\n" );
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document.write( "How many people in Ancient Greece do you think knew Geometry from Euclid's  books <BR>\n" );
document.write( "(in the volume as presented in his books) in the time of Euclid?<BR>\n" );
document.write( "I think such people can be counted on the fingers.\r<BR>\n" );
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document.write( "How many people knew Euclid's Geometry in the same volume in the Middle Ages? <BR>\n" );
document.write( "- Several hundred people out of many millions in Europe.\r<BR>\n" );
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document.write( "How many people now in the US know Geometry in the volume of Euclid's books? <BR>\n" );
document.write( "- Still the same several hundred out of 320 millions.\r<BR>\n" );
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document.write( "Nevertheless, we happily coexist.\r<BR>\n" );
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document.write( "Think about it.\r<BR>\n" );
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document.write( "And think when you post your comments.  May be, it is better do not post them, at all.\r<BR>\n" );
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document.write( "If you don't understand my solutions and explanations, consider to switch making embroidery using patterns.\r<BR>\n" );
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