Question 1208159
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Show that x^2 + 4 is {{{highlight(highlight(a))}}} prime.
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The complete formulation of the problem is


            Show that  x^2 + 4  is a prime polynomial

            over the set of polynomials with real coefficients.


        I will give two proofs.



<pre>
        First proof is very similar to that which I gave to other (similar) problem under this link
        <A 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> 


If x^2 + 4 is not a prime over the set of polynomials with real coefficients, 
then it is the product of two linear binomials

    x^2 + 4 = (ax+b)*(cx+d),


where a, b, c and d are real numbers.  Since ab must be equal to 1, the coefficients "a" and "b"
must be non-zero real numbers. But then the polynomial x^2 + 4 has two real roots

    {{{x[1]}}} = {{{-b/a}}}  and  {{{x[2]}}} = {{{-d/c}}}.


But, from the other hand, the discriminant of this quadratic polynomial is

    "b^2 - 4ac" = 0^2 - 4*1*4 = 0 - 16 = -16,

which is negative number.


A quadratic polynomial with negative discriminant can not have real roots.


This contradiction proves that given polynomial x^2 + 4 is a prime in the set of
polynomial with real coefficients.
</pre>

At this point, the first proof is completed.


<pre>
      Second proof is more simple.
      It does not use the conception of the discriminant.


If x^2 + 4 is not a prime over the set of polynomials with real coefficients, 
then it is the product of two linear binomials

    x^2 + 4 = (ax+b)*(cx+d),


where a, b, c and d are real numbers.  Since ab must be equal to 1, the coefficients "a" and "b"
must be non-zero real numbers. But then the polynomial x^2 + 4 has two real roots

    {{{x[1]}}} = {{{-b/a}}}  and  {{{x[2]}}} = {{{-d/c}}}.


From the other hand, it is easy to see that the quadratic function y = x^2 + 4
is always positive and, therefore, has no real roots.


This contradiction proves the statement and completes the solution.
</pre>

Solved/proved in full in two ways, for your better understanding.



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<U>Comment from student</U>: Well, easy for you. Difficult for most people.



<U>My response</U>: &nbsp;&nbsp;(1) &nbsp;&nbsp;Who authorized you to speak on behalf of the most people?

You yourself authorized yourself ? - Please get the lowest possible score from me for your comment.


(2) &nbsp;&nbsp;My goal is not to make it easy for all.

My goal is to make it educative for those who will read my post and is able to make 
minimal necessary mental efforts to understand my writing.


(3) &nbsp;&nbsp;And another question: &nbsp;for what reason do you come to this forum, 
if you do not want or are not able to make even minimal microscopic mental effort 
to absorb what was just explained to you?



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The idea that Mathematics should be understandable for everyone is simply crazy.
Just like the idea that Mathematics should be taught equally to everyone.


These ideas are of the same category as teaching everyone and everybody to play 
violin or to play piano to a master's level.


If you follow these ideas, you will have revolutions and civil wars in the country 
and in society every 20-30 years.


Mathematics at a good level should only be taught to those who have an inclination 
for it, who feel the harmony of mathematics and who are ready to make the necessary 
efforts to comprehend and master it.


The same as with music, with the piano and the violin.


For everyone else - the multiplication table, the square roots and a few of logarithms.


How many people in Ancient Greece do you think knew Geometry from Euclid's  books 
(in the volume as presented in his books) in the time of Euclid?
I think such people can be counted on the fingers.


How many people knew Euclid's Geometry in the same volume in the Middle Ages? 
- Several hundred people out of many millions in Europe.


How many people now in the US know Geometry in the volume of Euclid's books? 
- Still the same several hundred out of 320 millions.


Nevertheless, we happily coexist.


Think about it.


And think when you post your comments.  May be, it is better do not post them, at all.


If you don't understand my solutions and explanations, consider to switch making embroidery using patterns.