Answer :
To determine which expression is a prime polynomial, let's go through each option to see if it can be factored. A prime polynomial is a polynomial that cannot be factored into the product of two polynomials with integer coefficients.
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression can potentially be factored into polynomials of lower degree. After factoring, it will not remain in its original form if it is not prime.
B. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored because it has a common factor of 3. When factored, it becomes [tex]\(3(x^2 + 6y)\)[/tex], which means it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes, and it can be factored using the difference of cubes formula:
[tex]\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\][/tex]
In this case, let [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex]. Thus:
[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.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This expression has a common factor of [tex]\(x\)[/tex], allowing it to be factored out, making it non-prime.
[tex]\[(10x^3 - 5x^2 + 70x + 3)\][/tex]
Among the options, A remains unchanged as it cannot be factored further into polynomials with integer coefficients. Therefore, the prime polynomial is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression can potentially be factored into polynomials of lower degree. After factoring, it will not remain in its original form if it is not prime.
B. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored because it has a common factor of 3. When factored, it becomes [tex]\(3(x^2 + 6y)\)[/tex], which means it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes, and it can be factored using the difference of cubes formula:
[tex]\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\][/tex]
In this case, let [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex]. Thus:
[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.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This expression has a common factor of [tex]\(x\)[/tex], allowing it to be factored out, making it non-prime.
[tex]\[(10x^3 - 5x^2 + 70x + 3)\][/tex]
Among the options, A remains unchanged as it cannot be factored further into polynomials with integer coefficients. Therefore, the prime polynomial is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
