SOLUTION: Show that x^2 + x + 1 is prime.

Question 1207339: Show that x^2 + x + 1 is prime.
Answer by ikleyn(52805) (Show Source): You can put this solution on YOUR website!
.
Show that x^2 + x + 1 is prime.
~~~~~~~~~~~~~~~~~~~~~~~~

Consider the discriminant of this quadratic polynomial.

The discriminant is

    d = b^2 - 4ac = 1^2 - 4*1*1 = -3.


The discriminant is negative, which means that the polynomial does not have real roots.


If this polynomial be a composite over real numbers, it would be a product of two linear 
binomials (ax+b)*(cx+d).

But then it would have two real roots,  -b/a  and  -c/d.


Thus we get a CONTRADICTION with the fact proven above that this polynomial has no real roots.


The contradiction PROVES that the polynomial is not a composite over real numbers.


Hence, it is a PRIME over real numbers.

Solved.

RELATED QUESTIONS
Show that x^2 + 4 is prime. (answered by ikleyn,math_tutor2020)

Show that x^2 + 4 is prime. (answered by ikleyn)

Show that x^2 + 3 is prime. (answered by ikleyn)

Show that x^2 + 7x + 13 is prime. (answered by ikleyn)

Show that x^2 + 9 is prime. I will first list the pairs of integers whose product is... (answered by greenestamps,math_tutor2020)

is x^2 - x^35... (answered by solver91311)

Show that the rational function f(x)=x + 1/x -2 is continous at... (answered by tommyt3rd)

factor completely. If the polynomial is not factorable, write �prime.� x^2+7x+10... (answered by ankor@dixie-net.com)

factor, if the polynomial cant be factored then it is prime x^2+7x+10 please show... (answered by MathLover1)