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Answer :
To determine which expression is a prime polynomial, we need to understand what a prime polynomial is. A prime polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials with integer coefficients.
Let's examine each option to see if it can be factored:
A. [tex]\(3x^2 + 18y\)[/tex]
This polynomial can be factored. Both terms have a common factor of 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored further, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This resembles a difference of cubes, which follows the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this case:
[tex]\[ x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Let's look for common factors among the terms:
- The first three terms have a common factor of 5 and a factor of [tex]\(x^2\)[/tex] in the first two terms:
[tex]\[ 5x(2x^3 - x^2 + 14x) + 3x \][/tex]
Since it can potentially be factored further (or rewritten by factoring out common terms), it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's attempt to factor this polynomial. We can treat it as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[ (x^2 + 10)^2 - 10^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) \][/tex]
[tex]\[ (x^2)(x^2 + 20) \][/tex]
This is a disguisedly incomplete square trinomial. It doesn't appear as a standard prime polynomial. However, depending on further attempts at factorization over complex numbers or further insights into the factorization, we do not consider this a prime polynomial.
With the understanding of these steps, none of the given options is a prime polynomial, as they all can be factored. However, if a prime polynomial must be chosen from these options, the expected correct response is that none of these polynomials fits the criteria of being prime based on the given analysis.
Let's examine each option to see if it can be factored:
A. [tex]\(3x^2 + 18y\)[/tex]
This polynomial can be factored. Both terms have a common factor of 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored further, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This resembles a difference of cubes, which follows the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this case:
[tex]\[ x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Let's look for common factors among the terms:
- The first three terms have a common factor of 5 and a factor of [tex]\(x^2\)[/tex] in the first two terms:
[tex]\[ 5x(2x^3 - x^2 + 14x) + 3x \][/tex]
Since it can potentially be factored further (or rewritten by factoring out common terms), it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's attempt to factor this polynomial. We can treat it as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[ (x^2 + 10)^2 - 10^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) \][/tex]
[tex]\[ (x^2)(x^2 + 20) \][/tex]
This is a disguisedly incomplete square trinomial. It doesn't appear as a standard prime polynomial. However, depending on further attempts at factorization over complex numbers or further insights into the factorization, we do not consider this a prime polynomial.
With the understanding of these steps, none of the given options is a prime polynomial, as they all can be factored. However, if a prime polynomial must be chosen from these options, the expected correct response is that none of these polynomials fits the criteria of being prime based on the given analysis.
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