Answer :
Sure! Let's determine which expression is a prime polynomial by examining each option one by one. A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients.
Option A: [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor (GCF), which is 3:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
Option B: [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes. Recall the formula for factoring a difference of cubes:
[tex]\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\][/tex]
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], so:
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
Option C: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression looks like it could be a quadratic in terms of [tex]\(x^2\)[/tex]. We can try to factor it as though it was written in the form [tex]\(u^2 + 20u - 100\)[/tex] where [tex]\(u = x^2\)[/tex].
- Factoring gives:
[tex]\[
= (x^2 + 10)^2 - (10)^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) = (x^2)(x^2 + 20)
\][/tex]
- This expression can indeed be broken down, so it is not a prime polynomial.
Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can try factoring out the GCF, which is [tex]\(x\)[/tex]:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- Since we managed to factor out [tex]\(x\)[/tex], it is not a prime polynomial.
Among the given options, none of the polynomials are prime, since all can be factored further or simplified in some manner. It seems like there might have been a mistake, as the question typically expects only one option to be correct. If none match, it is always good to double-check the problem or explore within the context given, perhaps reaching out to a teacher or checking additional resources.
Option A: [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor (GCF), which is 3:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
Option B: [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes. Recall the formula for factoring a difference of cubes:
[tex]\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\][/tex]
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], so:
[tex]\[
x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, this is not a prime polynomial.
Option C: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression looks like it could be a quadratic in terms of [tex]\(x^2\)[/tex]. We can try to factor it as though it was written in the form [tex]\(u^2 + 20u - 100\)[/tex] where [tex]\(u = x^2\)[/tex].
- Factoring gives:
[tex]\[
= (x^2 + 10)^2 - (10)^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) = (x^2)(x^2 + 20)
\][/tex]
- This expression can indeed be broken down, so it is not a prime polynomial.
Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can try factoring out the GCF, which is [tex]\(x\)[/tex]:
[tex]\[
10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)
\][/tex]
- Since we managed to factor out [tex]\(x\)[/tex], it is not a prime polynomial.
Among the given options, none of the polynomials are prime, since all can be factored further or simplified in some manner. It seems like there might have been a mistake, as the question typically expects only one option to be correct. If none match, it is always good to double-check the problem or explore within the context given, perhaps reaching out to a teacher or checking additional resources.
