Answer :
To determine which expression is a prime polynomial, we need to analyze each expression one by one and test if they can be factored further. A prime polynomial is a polynomial that cannot be factored over the integers (at least one factor must be a constant).
Let's analyze each option:
Option 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]\(x^3\)[/tex] is [tex]\(a^3\)[/tex] and [tex]\(27y^6\)[/tex] is [tex]\((3y^2)^3\)[/tex], so it factors to [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
Option B: [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a common factor of 3: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored by taking out the greatest common factor, it is not a prime polynomial.
Option C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor out a common factor of [tex]\(x\)[/tex]: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since we can factor this expression further, it is not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial is a quadratic in form when we treat [tex]\(x^2\)[/tex] as a single variable: [tex]\((x^2)^2 + 20(x^2) - 100\)[/tex].
- We can attempt to factor it as [tex]\((x^2 + a)(x^2 + b)\)[/tex], but it does not factor neatly with integer coefficients.
- Thus, this polynomial is considered prime as it cannot be factored into polynomials with integer coefficients.
After analyzing each option, we find that Option D, [tex]\(x^4 + 20x^2 - 100\)[/tex], is a prime polynomial, as it cannot be factored further using integer coefficients.
Let's analyze each option:
Option 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]\(x^3\)[/tex] is [tex]\(a^3\)[/tex] and [tex]\(27y^6\)[/tex] is [tex]\((3y^2)^3\)[/tex], so it factors to [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
Option B: [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a common factor of 3: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored by taking out the greatest common factor, it is not a prime polynomial.
Option C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor out a common factor of [tex]\(x\)[/tex]: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since we can factor this expression further, it is not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial is a quadratic in form when we treat [tex]\(x^2\)[/tex] as a single variable: [tex]\((x^2)^2 + 20(x^2) - 100\)[/tex].
- We can attempt to factor it as [tex]\((x^2 + a)(x^2 + b)\)[/tex], but it does not factor neatly with integer coefficients.
- Thus, this polynomial is considered prime as it cannot be factored into polynomials with integer coefficients.
After analyzing each option, we find that Option D, [tex]\(x^4 + 20x^2 - 100\)[/tex], is a prime polynomial, as it cannot be factored further using integer coefficients.
