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Answer :
To determine which of the given polynomial expressions is a prime polynomial, we need to assess if each polynomial can be factored further, excluding the trivial case of multiplying by 1 or -1.
Let's evaluate each option one by one:
A. [tex]\(3x^2 + 18y\)[/tex]
This polynomial can be factored as:
[tex]\[3(x^2 + 6y)\][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial also factors because the terms have a common factor of [tex]\(x\)[/tex]:
[tex]\[x(10x^3 - 5x^2 + 70x + 3)\][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be rewritten and potentially factored. We notice:
[tex]\[x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100\][/tex]
However, this expression requires analysis or solving to check if it factors into simple binomial expressions. Following typical factor checks, we'll find that it can be factored using special methods or numerical approaches, so it's not necessarily prime.
D. [tex]\(x^3 - 27y^6\)[/tex]
This polynomial is a difference of cubes:
[tex]\[x^3 - (3y^2)^3\][/tex]
Expressing it using the difference of cubes formula:
[tex]\[a^3 - b^3 = (a-b)(a^2 + ab + b^2)\][/tex]
So, it factors into:
[tex]\[(x - 3y^2)(x^2 + 3xy^2 + 9y^4)\][/tex]
Since it also factors, it is not a prime polynomial.
Upon evaluating all the expressions, none of them qualify as prime polynomials because they can all be factored further.
Thus, the correct answer for a prime polynomial among the given choices is actually none.
Let's evaluate each option one by one:
A. [tex]\(3x^2 + 18y\)[/tex]
This polynomial can be factored as:
[tex]\[3(x^2 + 6y)\][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial also factors because the terms have a common factor of [tex]\(x\)[/tex]:
[tex]\[x(10x^3 - 5x^2 + 70x + 3)\][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be rewritten and potentially factored. We notice:
[tex]\[x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100\][/tex]
However, this expression requires analysis or solving to check if it factors into simple binomial expressions. Following typical factor checks, we'll find that it can be factored using special methods or numerical approaches, so it's not necessarily prime.
D. [tex]\(x^3 - 27y^6\)[/tex]
This polynomial is a difference of cubes:
[tex]\[x^3 - (3y^2)^3\][/tex]
Expressing it using the difference of cubes formula:
[tex]\[a^3 - b^3 = (a-b)(a^2 + ab + b^2)\][/tex]
So, it factors into:
[tex]\[(x - 3y^2)(x^2 + 3xy^2 + 9y^4)\][/tex]
Since it also factors, it is not a prime polynomial.
Upon evaluating all the expressions, none of them qualify as prime polynomials because they can all be factored further.
Thus, the correct answer for a prime polynomial among the given choices is actually none.
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