document.write( "Question 1208751: Show that x^2 + 3 is prime. \n" ); document.write( "

Algebra.Com's Answer #847187 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Show that x^2 + 3 is prime.
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document.write( "The precise meaning of this question is dark and unclear.\r\n" );
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document.write( "Therefore, I will re-formulate the request, to make it sensible.\r\n" );
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document.write( "    Show that polynomial  x^2 + 3  is a prime polynomial \r\n" );
document.write( "    in the ring of polynomials with real coefficients.\r\n" );
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document.write( "The solution is simple and beautiful, both at the same time.\r\n" );
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document.write( "Had the polynomial be non-prime (= composite) in this ring, \r\n" );
document.write( "it would be a product of two polynomials of degree 1, i.e. linear binomials\r\n" );
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document.write( "    x^2 + 3 = (ax+b)*(cx+d).\r\n" );
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document.write( "In this case, the polynomial would have real roots    and  .\r\n" );
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document.write( "But the polynomial  x^2 + 3  has always positive values for all real values of x,\r\n" );
document.write( "since x^2 is always non-negative, while the addend \"3\" is a positive constant.\r\n" );
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document.write( "This contradiction proves that the given polynomial is a prime polynomial \r\n" );
document.write( "in the ring of polynomials with real coefficients.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "The same reasoning works for polynomials with rational coefficients or/and for
\n" ); document.write( "polynomials with integer coefficients.\r
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\n" ); document.write( "\n" ); document.write( "I don't know what is your level in Math - therefore, I fully admit that you
\n" ); document.write( "may understand nothing from my explanation.\r
\n" ); document.write( "\n" ); document.write( "Why I may think so - because the problem in your post is worded mathematically incorrectly.\r
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\n" ); document.write( "\n" ); document.write( "On the other hand, since you brought this problem, I may assume that you have all
\n" ); document.write( "the necessary pre-requisites to understand my solution.\r
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\n" ); document.write( "\n" ); document.write( "The rest depends on you - not on me.\r
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\n" ); document.write( "\n" ); document.write( "With the full context this problem is for the first year undergraduate university
\n" ); document.write( "Math student, who recently started learning Abstract Algebra.\r
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\n" ); document.write( "\n" ); document.write( "Such a student should understand my solution adequately.\r
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