Answer :
To determine which expression is a prime polynomial, we need to check if any of these polynomials cannot be factored into simpler polynomials with integer coefficients.
Let's examine each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This is a difference of cubes, which can be factored as:
[tex]\[ x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's treat this as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100 \][/tex]
By trying to factor this similar to a quadratic, it's possible to find factors like:
[tex]\[ (x^2 + 10 + \sqrt{200})(x^2 + 10 - \sqrt{200}) \][/tex]
Because it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
First, we factor out the greatest common factor, which is [tex]\(x\)[/tex]:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be further factored (even by just factoring out common terms like [tex]\(x\)[/tex]), it is not a prime polynomial.
D. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor of 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
After examining all the options, none of these expressions are prime polynomials in the context of integer coefficients. This was a trick question since typically one would expect at least one polynomial to appear prime, but with these options, each can be factored using integer coefficients.
However, understanding the methods used to determine factorability is crucial for identifying rare cases where a polynomial remains prime. If there’s a misunderstanding, ensure to clarify which constraints define a polynomial as "prime" in your course or resource.
Let's examine each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This is a difference of cubes, which can be factored as:
[tex]\[ x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's treat this as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100 \][/tex]
By trying to factor this similar to a quadratic, it's possible to find factors like:
[tex]\[ (x^2 + 10 + \sqrt{200})(x^2 + 10 - \sqrt{200}) \][/tex]
Because it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
First, we factor out the greatest common factor, which is [tex]\(x\)[/tex]:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be further factored (even by just factoring out common terms like [tex]\(x\)[/tex]), it is not a prime polynomial.
D. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor of 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
After examining all the options, none of these expressions are prime polynomials in the context of integer coefficients. This was a trick question since typically one would expect at least one polynomial to appear prime, but with these options, each can be factored using integer coefficients.
However, understanding the methods used to determine factorability is crucial for identifying rare cases where a polynomial remains prime. If there’s a misunderstanding, ensure to clarify which constraints define a polynomial as "prime" in your course or resource.
