Answer :
To determine which expression is a prime polynomial, let's review the options and analyze each one:
1. Expression A: [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a common factor of 3 from each term:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
- Since it can be factored further, this is not a prime polynomial.
2. Expression B: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Notice that each term has an [tex]\(x\)[/tex], so we can factor that out:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
3. Expression C: [tex]\(x^3 - 27y^6\)[/tex]
- This is a difference of cubes, as [tex]\(x^3 - (3y^2)^3\)[/tex].
- It can be factored using the formula for the difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
- Applying the formula:
[tex]\[
x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
4. Expression D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial doesn't have an obvious factorization pattern like the other expressions. It can't be factored into the product of two non-constant polynomials using common factorization techniques.
- While further specific testing (or more advanced factorization methods) might be needed to definitively prove its primacy, at a basic level of examination using typical methods, it appears to be a prime polynomial.
Therefore, the expression that is a prime polynomial is D. [tex]\(x^4 + 20x^2 - 100\)[/tex].
1. Expression A: [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a common factor of 3 from each term:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
- Since it can be factored further, this is not a prime polynomial.
2. Expression B: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Notice that each term has an [tex]\(x\)[/tex], so we can factor that out:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
3. Expression C: [tex]\(x^3 - 27y^6\)[/tex]
- This is a difference of cubes, as [tex]\(x^3 - (3y^2)^3\)[/tex].
- It can be factored using the formula for the difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
- Applying the formula:
[tex]\[
x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
4. Expression D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial doesn't have an obvious factorization pattern like the other expressions. It can't be factored into the product of two non-constant polynomials using common factorization techniques.
- While further specific testing (or more advanced factorization methods) might be needed to definitively prove its primacy, at a basic level of examination using typical methods, it appears to be a prime polynomial.
Therefore, the expression that is a prime polynomial is D. [tex]\(x^4 + 20x^2 - 100\)[/tex].
