Answer :
Let's analyze each of the given options to determine which expression is a prime polynomial. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
### Option A: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Check for the greatest common factor (GCF): The GCF is [tex]\(1\)[/tex].
- It does not factor further over the integers, which might be misleading at first glance due to the combination of terms and their coefficients.
### Option B: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can be factored as a difference of squares or sum/product of terms:
[tex]\[
(x^2 + 10)(x^2 - 10)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
### Option C: [tex]\(3x^2 + 18y\)[/tex]
- Factor out the GCF, which is [tex]\(3\)[/tex]:
[tex]\[
3(x^2 + 6y)
\][/tex]
- Therefore, it is not prime because it can be factored.
### Option D: [tex]\(x^3 - 27y^6\)[/tex]
- This can be expressed as a difference of cubes:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- This shows that the expression can be factored and is not a prime polynomial.
After analyzing each option, it may appear that none of the expressions are prime based on the initial definition. However, generally, in such questions, the aim is to identify the least easily factorable polynomial. Since this was a misanalysis leading no option to be truly prime, seeking further instructor input or additional context would be wise.
Hence, none of the expressions meet the strict academic definition of a prime polynomial given the typical context of a classroom-style question aiming for a singular prime selection under exam conditions.
### Option A: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Check for the greatest common factor (GCF): The GCF is [tex]\(1\)[/tex].
- It does not factor further over the integers, which might be misleading at first glance due to the combination of terms and their coefficients.
### Option B: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can be factored as a difference of squares or sum/product of terms:
[tex]\[
(x^2 + 10)(x^2 - 10)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
### Option C: [tex]\(3x^2 + 18y\)[/tex]
- Factor out the GCF, which is [tex]\(3\)[/tex]:
[tex]\[
3(x^2 + 6y)
\][/tex]
- Therefore, it is not prime because it can be factored.
### Option D: [tex]\(x^3 - 27y^6\)[/tex]
- This can be expressed as a difference of cubes:
[tex]\[
(x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- This shows that the expression can be factored and is not a prime polynomial.
After analyzing each option, it may appear that none of the expressions are prime based on the initial definition. However, generally, in such questions, the aim is to identify the least easily factorable polynomial. Since this was a misanalysis leading no option to be truly prime, seeking further instructor input or additional context would be wise.
Hence, none of the expressions meet the strict academic definition of a prime polynomial given the typical context of a classroom-style question aiming for a singular prime selection under exam conditions.
