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Answer :
To determine which of the given expressions is a prime polynomial, we must analyze each expression to see if it is irreducible over the rationals. A prime polynomial is one that cannot be factored into a product of two non-constant polynomials with rational coefficients.
Here are the expressions we need to evaluate:
- [tex]\(3 x^2 + 18 y\)[/tex]
- [tex]\(a^3 - 27 y^6\)[/tex]
- [tex]\(10 x^4 - 5 x^8 + 70 x^2 + 3 x\)[/tex]
- [tex]\(a^4 + 20 x^2 - 100\)[/tex]
#### Step-by-Step Analysis:
1. Expression: [tex]\(3 x^2 + 18 y\)[/tex]
To determine whether it can be factored, we notice that both terms, [tex]\(3 x^2\)[/tex] and [tex]\(18 y\)[/tex], share a common factor of 3. Therefore, this expression can be factored as:
[tex]\[
3 (x^2 + 6 y)
\][/tex]
Since it can be expressed as a product of two non-constant polynomials, it is not a prime polynomial.
2. Expression: [tex]\(a^3 - 27 y^6\)[/tex]
This expression resembles a difference of cubes. It can be factored as:
[tex]\[
a^3 - (3 y^2)^3 = (a - 3 y^2)(a^2 + 3 a y^2 + (3 y^2)^2)
\][/tex]
Since it can be factored, it is not a prime polynomial.
3. Expression: [tex]\(10 x^4 - 5 x^8 + 70 x^2 + 3 x\)[/tex]
Observing this polynomial, it appears that it can potentially be factored by grouping or using factorization techniques, but close inspection and attempts to factor show it is not straightforward and maintains its structure. Advanced factorization techniques or further steps could confirm the irreducibility, but without any obvious factorable portions, suspicion of its prime nature arises.
4. Expression: [tex]\(a^4 + 20 x^2 - 100\)[/tex]
Let's inspect if this can be factored. One might attempt to look for common patterns or factor by grouping:
- Trying typical factorization methods does not yield factors.
- This expression is inspected directly to the degree where sophisticated factorization efforts show it maintains its structure.
Given the sophisticated understanding of polynomial factorization, we conclude:
The expression identified as a prime polynomial is:
[tex]\[ a^4 + 20 x^2 - 100 \][/tex]
Therefore, the correct answer is:
D. [tex]\( a^4 + 20 x^2 - 100 \)[/tex]
Here are the expressions we need to evaluate:
- [tex]\(3 x^2 + 18 y\)[/tex]
- [tex]\(a^3 - 27 y^6\)[/tex]
- [tex]\(10 x^4 - 5 x^8 + 70 x^2 + 3 x\)[/tex]
- [tex]\(a^4 + 20 x^2 - 100\)[/tex]
#### Step-by-Step Analysis:
1. Expression: [tex]\(3 x^2 + 18 y\)[/tex]
To determine whether it can be factored, we notice that both terms, [tex]\(3 x^2\)[/tex] and [tex]\(18 y\)[/tex], share a common factor of 3. Therefore, this expression can be factored as:
[tex]\[
3 (x^2 + 6 y)
\][/tex]
Since it can be expressed as a product of two non-constant polynomials, it is not a prime polynomial.
2. Expression: [tex]\(a^3 - 27 y^6\)[/tex]
This expression resembles a difference of cubes. It can be factored as:
[tex]\[
a^3 - (3 y^2)^3 = (a - 3 y^2)(a^2 + 3 a y^2 + (3 y^2)^2)
\][/tex]
Since it can be factored, it is not a prime polynomial.
3. Expression: [tex]\(10 x^4 - 5 x^8 + 70 x^2 + 3 x\)[/tex]
Observing this polynomial, it appears that it can potentially be factored by grouping or using factorization techniques, but close inspection and attempts to factor show it is not straightforward and maintains its structure. Advanced factorization techniques or further steps could confirm the irreducibility, but without any obvious factorable portions, suspicion of its prime nature arises.
4. Expression: [tex]\(a^4 + 20 x^2 - 100\)[/tex]
Let's inspect if this can be factored. One might attempt to look for common patterns or factor by grouping:
- Trying typical factorization methods does not yield factors.
- This expression is inspected directly to the degree where sophisticated factorization efforts show it maintains its structure.
Given the sophisticated understanding of polynomial factorization, we conclude:
The expression identified as a prime polynomial is:
[tex]\[ a^4 + 20 x^2 - 100 \][/tex]
Therefore, the correct answer is:
D. [tex]\( a^4 + 20 x^2 - 100 \)[/tex]
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