Respuesta :
You can use the definition of prime polynomials to find out which polynomial is prime and which is not.
The prime polynomials in the given options are:
Option 1: [tex]2x^2 + 7x + 1[/tex]
Option 3: [tex]3x^2 + 8[/tex]
What are prime polynomials?
Those polynomials with integer coefficients that cannot be factored further, with factors of lower degree and integer coefficients are called prime polynomial.
(it is necessary that no factors exists having their coefficients are still integers and they're of lower degree)
Which of the given polynomials are prime?
Option 1: [tex]2x^2 + 7x + 1[/tex]
We have to check it manually. The roots of this equation are
not integer or integer fraction (checked from calculator), and does't factor into smaller integer coefficient polynomials.
(you can check if its getting factored, manually, and most of the times, if it is not prime, it will get factored).
Thus, it is a prime polynomial.
Option 2: [tex]5x^2 - 10x +5[/tex]
We can rewrite it as
[tex]5x^2 -10x + 5\\5x^2 - 5x -5x =5\\5x(x-1) -5(x-1)\\(5x-5)(x-1)[/tex]
Thus, it is possible to write it as product of lower degree polynomial with integer coefficients, thus it is not a prime polynomial.
Option 3: [tex]3x^2 + 8[/tex]
Since 3 is prime number and since given polynomial is quadratic thus lower degree would be only linear, thus, let
[tex](3x+a)(x+b) = 3x^2 + 8\\3x^2 + (a+3b)x + ab = 3x^2 + 8\\a = -3b\\ab = 8\\(-3b)b = 8\\b = \sqrt{-8/3}[/tex]
b is not a real number, so the factorization is not polynomial factorization.
Thus, it is prime polynomial.
Option 4: [tex]4x^2 - 25[/tex]
We can rewrite it as:
[tex](2x)^2 - 5^2= (2x+5)(2x-5)[/tex]
Thus, not a prime polynomial.
Option 5: [tex]x^2 - 36\\[/tex]
We can rewrite it as:
[tex]x^2 - 6^2 = (x-6)(x+6)[/tex]
Thus, not a prime polynomial.
Thus,
The prime polynomials in the given options are:
Option 1: [tex]2x^2 + 7x + 1[/tex]
Option 3: [tex]3x^2 + 8[/tex]
Learn more about prime polynomials here:
https://brainly.com/question/10717989
