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Answer :
To determine which expression is a prime polynomial, we need to check if any of the given polynomials can be further factored. A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients.
Let's examine each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This expression is actually a difference of cubes:
[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.
B. [tex]\(3x^2 + 18y\)[/tex]
We can factor out the greatest common factor:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, this is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression can be thought of as a quadratic in form once we let [tex]\( u = x^2 \)[/tex]:
[tex]\[ u^2 + 20u - 100 = (u + 10)(u - 10) = (x^2 + 10)(x^2 - 10) \][/tex]
Thus, it can be factored, so it is not a prime polynomial.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
It's not straightforwardly factorable by common methods. The terms do not share a common factor other than 1, and attempts to factor it further do not succeed in breaking it into simpler polynomials with integer coefficients, so this polynomial cannot be factored further.
Therefore, the prime polynomial is [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex].
Let's examine each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This expression is actually a difference of cubes:
[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.
B. [tex]\(3x^2 + 18y\)[/tex]
We can factor out the greatest common factor:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, this is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression can be thought of as a quadratic in form once we let [tex]\( u = x^2 \)[/tex]:
[tex]\[ u^2 + 20u - 100 = (u + 10)(u - 10) = (x^2 + 10)(x^2 - 10) \][/tex]
Thus, it can be factored, so it is not a prime polynomial.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
It's not straightforwardly factorable by common methods. The terms do not share a common factor other than 1, and attempts to factor it further do not succeed in breaking it into simpler polynomials with integer coefficients, so this polynomial cannot be factored further.
Therefore, the prime polynomial is [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex].
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