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Answer :
To determine which expression is a prime polynomial, let's analyze the options one by one. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with integer coefficients.
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]\( x^3 - (3y^2)^3 \)[/tex] can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This expression can be treated as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let [tex]\( z = x^2 \)[/tex]:
[tex]\[ z^2 + 20z - 100 \][/tex]
To determine if it can be factored further, we check for possible factorization, such as using the quadratic formula. However, since it simplifies into factors like:
[tex]\[ (x^2 + 10 - \sqrt{200})(x^2 + 10 + \sqrt{200}) \][/tex]
Here, it indicates factoring is not straightforward, yet it suggests potential factors, thus it might not be completely prime.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This expression appears complex for quick factorization. Upon analysis of polynomial factorization techniques or trying known methods like grouping and long division, we realize there's no obvious factorization evident with integer coefficients. Therefore, we conclude it cannot be factored further and is considered a prime polynomial.
D. [tex]\( 3x^2 + 18y \)[/tex]
This expression is linear in terms of [tex]\( y \)[/tex] and can be factored as:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since factoring is possible, it is not a prime polynomial.
After evaluating each expression, the expression [tex]\( C: 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex] is a prime polynomial because it cannot be factored further with integer coefficients. Thus, the correct answer is option C.
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]\( x^3 - (3y^2)^3 \)[/tex] can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This expression can be treated as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let [tex]\( z = x^2 \)[/tex]:
[tex]\[ z^2 + 20z - 100 \][/tex]
To determine if it can be factored further, we check for possible factorization, such as using the quadratic formula. However, since it simplifies into factors like:
[tex]\[ (x^2 + 10 - \sqrt{200})(x^2 + 10 + \sqrt{200}) \][/tex]
Here, it indicates factoring is not straightforward, yet it suggests potential factors, thus it might not be completely prime.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This expression appears complex for quick factorization. Upon analysis of polynomial factorization techniques or trying known methods like grouping and long division, we realize there's no obvious factorization evident with integer coefficients. Therefore, we conclude it cannot be factored further and is considered a prime polynomial.
D. [tex]\( 3x^2 + 18y \)[/tex]
This expression is linear in terms of [tex]\( y \)[/tex] and can be factored as:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since factoring is possible, it is not a prime polynomial.
After evaluating each expression, the expression [tex]\( C: 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex] is a prime polynomial because it cannot be factored further with integer coefficients. Thus, the correct answer is option C.
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