Answer :
To determine which expression is a prime polynomial, we need to understand that a prime polynomial is one that cannot be factored into the product of two non-constant polynomials.
Let's analyze each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial has a common factor of [tex]\(x\)[/tex]. It can be factored as [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex], so it can be factored further. Therefore, it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial has a common factor of [tex]\(3\)[/tex]. It can be factored as [tex]\(3(x^2 + 6y)\)[/tex], indicating it can be broken down into simpler polynomials. Thus, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial can be explored for potential factoring. It can be expressed as a product of polynomials like [tex]\((x^2 + 10)(x^2 - 10)\)[/tex], so it is not a prime polynomial because it can be factored further.
D. [tex]\(x^8 - 27y^6\)[/tex]
- This expression suggests the use of algebraic identities, such as the sum or difference of powers. It can be recognized and factored as [tex]\((x^4)^2 - (3y^2)^3\)[/tex], which can be further explored for factorization. Therefore, it is not a prime polynomial.
After assessing each option, we can conclude that none of the given expressions fit the definition of a prime polynomial, as they all can be factored into simpler polynomials. Hence, none of them is a prime polynomial.
Let's analyze each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial has a common factor of [tex]\(x\)[/tex]. It can be factored as [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex], so it can be factored further. Therefore, it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial has a common factor of [tex]\(3\)[/tex]. It can be factored as [tex]\(3(x^2 + 6y)\)[/tex], indicating it can be broken down into simpler polynomials. Thus, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial can be explored for potential factoring. It can be expressed as a product of polynomials like [tex]\((x^2 + 10)(x^2 - 10)\)[/tex], so it is not a prime polynomial because it can be factored further.
D. [tex]\(x^8 - 27y^6\)[/tex]
- This expression suggests the use of algebraic identities, such as the sum or difference of powers. It can be recognized and factored as [tex]\((x^4)^2 - (3y^2)^3\)[/tex], which can be further explored for factorization. Therefore, it is not a prime polynomial.
After assessing each option, we can conclude that none of the given expressions fit the definition of a prime polynomial, as they all can be factored into simpler polynomials. Hence, none of them is a prime polynomial.
