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Answer :
Let's discuss each option to determine which one is a prime polynomial. A prime polynomial is one that cannot be factored into polynomials of lower degrees with integer coefficients.
A. [tex]\( x^3 - 27y^6 \)[/tex]
This expression is recognized as a difference of cubes, which can be factored using the formula for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this case, [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex], so it can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
This means it is not a prime polynomial.
B. [tex]\( 3x^2 + 18y \)[/tex]
Here, you can factor out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
This factoring shows it is not a prime polynomial.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
At a glance, this polynomial doesn't have an obvious factor that can be factored using integer coefficients. Although it's more complex, it appears least reducible with immediate methods commonly taught in algebra for factoring over integers.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This expression can be treated as a quadratic in form by letting [tex]\( z = x^2 \)[/tex]:
[tex]\[ z^2 + 20z - 100 \][/tex]
This polynomial can be factored using the quadratic formula or other methods, indicating it is not a prime polynomial.
Finally, among the options, Option C is the one where further factoring over integers is not readily apparent. Thus, Option C is identified as the prime polynomial based on typical analysis criteria over integers.
A. [tex]\( x^3 - 27y^6 \)[/tex]
This expression is recognized as a difference of cubes, which can be factored using the formula for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this case, [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex], so it can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
This means it is not a prime polynomial.
B. [tex]\( 3x^2 + 18y \)[/tex]
Here, you can factor out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
This factoring shows it is not a prime polynomial.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
At a glance, this polynomial doesn't have an obvious factor that can be factored using integer coefficients. Although it's more complex, it appears least reducible with immediate methods commonly taught in algebra for factoring over integers.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This expression can be treated as a quadratic in form by letting [tex]\( z = x^2 \)[/tex]:
[tex]\[ z^2 + 20z - 100 \][/tex]
This polynomial can be factored using the quadratic formula or other methods, indicating it is not a prime polynomial.
Finally, among the options, Option C is the one where further factoring over integers is not readily apparent. Thus, Option C is identified as the prime polynomial based on typical analysis criteria over integers.
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