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The polynomial that is prime is given by Option C: [tex]10x^2 - 4x +3x +6[/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)
Checking all the listed polynomials for them being prime or not:
- Case 1: [tex]x^3+3x^2+2x+6[/tex]
Rewriting it as:
[tex]x^3+3x^2+2x+6 = x^2(x+3) + 2(x+3) = (x^2 + 2)(x+3)[/tex]
shows that this polynomial has factors. Thus, it isn't prime polynomial
- Case 2: [tex]x^3 + 3x^2 -2x -6[/tex]
Rewriting it as:
[tex]x^3 + 3x^2 -2x -6 = x^2(x+3) -2(x+3) = (x^2-2)(x+3)[/tex]
shows that this polynomial has factors. Thus, it isn't prime polynomial
- Case 3: [tex]10x^2 - 4x +3x +6[/tex]
We can rewrite it as:
[tex]10x^2 -4x + 3x -6 = 10x^2 -x -6 = f(x) \: \rm (say)[/tex]
This polynomial is prime polynomial as it is irreducible in terms of polynomials having their coefficients as integers.
Case 4: [tex]10x^2 -10x + 6x - 6[/tex]
It can be factored as:
[tex]10x(x-1) + 6(x-1) = (x-1)(10x-6)[/tex]
This is not a prime polynomial.
Thus, the polynomial that is prime is given by Option C: [tex]10x^2 - 4x +3x +6[/tex]
Learn more about prime polynomials here:
https://brainly.com/question/10717989
Answer:
The polynomial that is prime is given by Option C:
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)
Checking all the listed polynomials for them being prime or not:
Case 1:
Rewriting it as:
shows that this polynomial has factors. Thus, it isn't prime polynomial
Case 2:
Rewriting it as:
shows that this polynomial has factors. Thus, it isn't prime polynomial
Case 3:
We can rewrite it as:
This polynomial is prime polynomial as it is irreducible in terms of polynomials having their coefficients as integers.
Case 4:
It can be factored as:
This is not a prime polynomial.
Thus, the polynomial that is prime is given by Option C:
Learn more about prime polynomials here:
brainly.com/question/10717989
Step-by-step explanation:
