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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 polynomials of lower degree with integer coefficients.
Let's evaluate each option:
Option A: [tex]\(x^3 - 27y^6\)[/tex]
This expression can be factored using the difference of cubes formula:
[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, it is not a prime polynomial.
Option B: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
First, we check for common factors:
We can factor an [tex]\(x\)[/tex] out:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Further factorization can be complex, but factorizable factors can potentially exist, making it non-prime.
Option C: [tex]\(3x^2 + 18y\)[/tex]
This polynomial also has a common factor:
You can factor out a 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored further, it is not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be factored as follows:
This can be seen as a difference of squares by rewriting it:
[tex]\[ (x^2 + 10)^2 - 10^2 \][/tex]
Factoring further, it becomes:
[tex]\[ (x^2 + 20)(x^2 - 5) \][/tex]
Hence, it can be factored and is not a prime polynomial.
After analyzing each option, we find that none of the given polynomials are prime because they can all be factored into polynomials of lower degrees. Therefore, none of the options A, B, C, or D is a prime polynomial.
Let's evaluate each option:
Option A: [tex]\(x^3 - 27y^6\)[/tex]
This expression can be factored using the difference of cubes formula:
[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, it is not a prime polynomial.
Option B: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
First, we check for common factors:
We can factor an [tex]\(x\)[/tex] out:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Further factorization can be complex, but factorizable factors can potentially exist, making it non-prime.
Option C: [tex]\(3x^2 + 18y\)[/tex]
This polynomial also has a common factor:
You can factor out a 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored further, it is not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be factored as follows:
This can be seen as a difference of squares by rewriting it:
[tex]\[ (x^2 + 10)^2 - 10^2 \][/tex]
Factoring further, it becomes:
[tex]\[ (x^2 + 20)(x^2 - 5) \][/tex]
Hence, it can be factored and is not a prime polynomial.
After analyzing each option, we find that none of the given polynomials are prime because they can all be factored into polynomials of lower degrees. Therefore, none of the options A, B, C, or D is a prime polynomial.
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