Answer :
To determine which expression is a prime polynomial, let's look at each option and check whether it can be factored:
A. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the common factor, which is 3. So, it becomes:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes, which can be factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex]. Applying the formula, we have:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This expression can also be partially factored by taking out the common factor, which is [tex]\(x\)[/tex]. So, it becomes:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial, in this form, does not factor into lower-degree polynomials over integers or any obvious factorable terms.
After checking each option, it turns out that the expression [tex]\(x^4 + 20x^2 - 100\)[/tex] (Option D) does not factor any further over the integers and is considered a prime polynomial. Therefore, the correct answer is:
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex] is a prime polynomial.
A. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the common factor, which is 3. So, it becomes:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes, which can be factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex]. Applying the formula, we have:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This expression can also be partially factored by taking out the common factor, which is [tex]\(x\)[/tex]. So, it becomes:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial, in this form, does not factor into lower-degree polynomials over integers or any obvious factorable terms.
After checking each option, it turns out that the expression [tex]\(x^4 + 20x^2 - 100\)[/tex] (Option D) does not factor any further over the integers and is considered a prime polynomial. Therefore, the correct answer is:
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex] is a prime polynomial.
