Answer :
To determine which expression is a prime polynomial, we need to identify which of them cannot be factored into polynomials of lower degree.
Let's go through each expression:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This expression is not a prime polynomial. It can be factored by pulling out a common factor, [tex]\(x\)[/tex], which gives [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. This means it can be simplified and thus is not prime.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes, which can be factored using the identity [tex]\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)[/tex]. In this case, [tex]\(x^3 - (3y^2)^3\)[/tex] can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex]. Therefore, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression cannot be factored into polynomials of lower degree over the integers, which makes it a prime polynomial. Since no simpler polynomial factorization exists for it, it is considered prime.
D. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor, which is 3, resulting in [tex]\(3(x^2 + 6y)\)[/tex]. Because it can be factored further, it is not a prime polynomial.
Hence, the correct answer is C. [tex]\(x^4 + 20x^2 - 100\)[/tex], as it is the prime polynomial.
Let's go through each expression:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This expression is not a prime polynomial. It can be factored by pulling out a common factor, [tex]\(x\)[/tex], which gives [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. This means it can be simplified and thus is not prime.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes, which can be factored using the identity [tex]\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)[/tex]. In this case, [tex]\(x^3 - (3y^2)^3\)[/tex] can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex]. Therefore, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression cannot be factored into polynomials of lower degree over the integers, which makes it a prime polynomial. Since no simpler polynomial factorization exists for it, it is considered prime.
D. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor, which is 3, resulting in [tex]\(3(x^2 + 6y)\)[/tex]. Because it can be factored further, it is not a prime polynomial.
Hence, the correct answer is C. [tex]\(x^4 + 20x^2 - 100\)[/tex], as it is the prime polynomial.
