Question 1208160
.


The correct formulation of the problem is


            Prove that  x^2 + 5x + 7  is a prime polynomial.



<pre>
The  {{{highlight(highlight(TRUE))}}}  meaning of this assignment is to prove that this given polynomial is a prime 
over real coefficients polynomials.  Again: not over the set of polynomial with integer coefficients,
but over much wider set of polynomials with real coefficients.


If x^2 + 5x + 7 is not a prime over the set of polynomials with real coefficients, 
then it is a product of two linear binomials

    x^2 + 5x + 7 = (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 + 5x + 7 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 " = 5^2 - 4*1*7 = 25 - 28 = -3,

which is negative number.


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


This contradiction proves that given polynomial x^2 + 5x + 7 is a prime in the set of
polynomial with real coefficients.
</pre>

At this point, the proof is completed and the problem is solved in full.


You should clearly understand what is the difference in the formulation of the problem
and in its solution between my post and the post by Edwin.


This difference is a huge.



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From this solution, one can formulate this simple criterion for a quadratic polynomial
with real coefficients to be a prime.  The necessary and sufficient condition for it
is that this quadratic polynomial has negative discriminant.



/////////////////////



In response to Edwin's comment.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Edwin, &nbsp;I think that average middle school student don't know these words &nbsp;" prime polynomial " 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and do not understand at all, &nbsp;what these words really mean, &nbsp;unless this student has an additional training 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;from outside, &nbsp;like a &nbsp;Math school, &nbsp;or a &nbsp;Math circle, &nbsp;or a personal &nbsp;Math teacher/trainer. 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, &nbsp;I think that my solution is adequate and adequately targeted.



After your comment, &nbsp;Edwin, &nbsp;I searched Internet for key words &nbsp;" proving that a polynomial is prime ",

and first link from &nbsp;Google was &nbsp;THIS


https://virtualnerd.com/algebra-2/polynomials/equations/factoring-strategies/prime-polynomial-definition


which is, &nbsp;OBVIOUSLY, &nbsp;at least &nbsp;Algebra-II &nbsp;level.



For us, &nbsp;adult persons of the age &nbsp;60 - 70 - 80, &nbsp;graduated in &nbsp;Math from universities in &nbsp;60-ies and &nbsp;70-ies, 
these notions/conceptions &nbsp;" factoring polynomials " &nbsp;and &nbsp;" prime polynomials " &nbsp;look like very close.
But for young students, &nbsp;these notions/conceptions are separated by &nbsp;3-5 years distance, &nbsp;at least.


In order to understand, &nbsp;what is a &nbsp;" prime polynomial ", &nbsp;a student should know/absorb 
the notions &nbsp;" a field " &nbsp;and &nbsp;" a ring " &nbsp;from abstract &nbsp;Algebra, &nbsp;which is the level 
of &nbsp;introductory undergraduate &nbsp;Math courses of college and/or university, &nbsp;in reality.



Edwin, I also looked into the reference in your post.


To me, it is bad style of teaching (anti-pedagogic), when the question asks about a notion/conception,
(a prime polynomial), which was not defined and explained previously.


If this author wants to be in good style and proposes this problem to 6-th grade students,
he/she must ask about factoring, not about "a prime polynomial".



And one more, &nbsp;probably, &nbsp;the last my note.


Edwin, &nbsp;why do you refer to other posts in the Internet to disprove my solution?


Do you think that they are more authoritative than my posts ?



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It even does not come to my mind.



On the contrary, &nbsp;it seems to me that each my post should be carved 
in stone and installed near each school for students and teachers to learn.


And in the dark hours of the day they should be additionally illuminated for better visibility.