Answer :
To determine which expression is a prime polynomial, we need to understand what a prime polynomial is. A polynomial is considered prime if it cannot be factored into the product of two or more non-constant polynomials over the set of integers.
Let's examine each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial might look complex, but it is actually prime in terms of factoring over integers. There is no way to break it down into simpler polynomials with integer coefficients.
B. [tex]\(10x^4 - 5x^8 + 70x^2 + 3x\)[/tex]
- This expression has multiple terms and higher powers, suggesting it might be factorable. In fact, it can be rewritten and factored in such a way that it is not a prime polynomial.
C. [tex]\(x^8 - 27y^5\)[/tex]
- This is a binomial with a power of 8 and 5. While it may seem complex, considering its structure, it falls under a form that can be decomposed using special binomial factorization methods.
D. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by taking out the greatest common factor of the terms, which is 3. This means it’s not prime as it can be factored into simpler terms.
So, after carefully evaluating each option, the polynomial that is considered prime, meaning it cannot be factored further into simpler polynomials with integer coefficients, is:
Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
This is consistent with evaluating the expressions properly according to the standards of prime polynomials.
Let's examine each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial might look complex, but it is actually prime in terms of factoring over integers. There is no way to break it down into simpler polynomials with integer coefficients.
B. [tex]\(10x^4 - 5x^8 + 70x^2 + 3x\)[/tex]
- This expression has multiple terms and higher powers, suggesting it might be factorable. In fact, it can be rewritten and factored in such a way that it is not a prime polynomial.
C. [tex]\(x^8 - 27y^5\)[/tex]
- This is a binomial with a power of 8 and 5. While it may seem complex, considering its structure, it falls under a form that can be decomposed using special binomial factorization methods.
D. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by taking out the greatest common factor of the terms, which is 3. This means it’s not prime as it can be factored into simpler terms.
So, after carefully evaluating each option, the polynomial that is considered prime, meaning it cannot be factored further into simpler polynomials with integer coefficients, is:
Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
This is consistent with evaluating the expressions properly according to the standards of prime polynomials.
