Answer :
To determine which expression is a prime polynomial, we need to check each option to see if it can be factored further. A prime polynomial cannot be factored into the product of two non-constant polynomials with real coefficients.
Let's analyze each option:
Option A: [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes, as it can be written as [tex]\(x^3 - (3y^2)^3\)[/tex].
- A difference of cubes has the form [tex]\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)[/tex].
- Applying this formula, [tex]\(x^3 - (3y^2)^3\)[/tex] can be factored into [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Therefore, [tex]\(x^3 - 27y^6\)[/tex] is not a prime polynomial because it can be factored.
Option B: [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by pulling out the greatest common factor, which is 3.
- Factoring gives us [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, [tex]\(3x^2 + 18y\)[/tex] is not a prime polynomial.
Option C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor out the common factor [tex]\(x\)[/tex], resulting in [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Even though the inner polynomial [tex]\(10x^3 - 5x^2 + 70x + 3\)[/tex] may seem complicated for further factoring with simple methods, the expression is not in its simplest prime polynomial form since we extracted an [tex]\(x\)[/tex].
- Therefore, [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex] is not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- Attempts to factor this by grouping or other standard algebraic methods do not quickly reveal further factorization.
- There is no straightforward factor visible, suggesting it cannot be easily divided into the product of simpler polynomials with real coefficients.
- Therefore, this polynomial fits the definition of being prime, meaning it cannot be factored further using standard algebraic techniques.
So, the expression that is a prime polynomial is Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex].
Let's analyze each option:
Option A: [tex]\(x^3 - 27y^6\)[/tex]
- This expression is a difference of cubes, as it can be written as [tex]\(x^3 - (3y^2)^3\)[/tex].
- A difference of cubes has the form [tex]\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)[/tex].
- Applying this formula, [tex]\(x^3 - (3y^2)^3\)[/tex] can be factored into [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Therefore, [tex]\(x^3 - 27y^6\)[/tex] is not a prime polynomial because it can be factored.
Option B: [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by pulling out the greatest common factor, which is 3.
- Factoring gives us [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, [tex]\(3x^2 + 18y\)[/tex] is not a prime polynomial.
Option C: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor out the common factor [tex]\(x\)[/tex], resulting in [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Even though the inner polynomial [tex]\(10x^3 - 5x^2 + 70x + 3\)[/tex] may seem complicated for further factoring with simple methods, the expression is not in its simplest prime polynomial form since we extracted an [tex]\(x\)[/tex].
- Therefore, [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex] is not a prime polynomial.
Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- Attempts to factor this by grouping or other standard algebraic methods do not quickly reveal further factorization.
- There is no straightforward factor visible, suggesting it cannot be easily divided into the product of simpler polynomials with real coefficients.
- Therefore, this polynomial fits the definition of being prime, meaning it cannot be factored further using standard algebraic techniques.
So, the expression that is a prime polynomial is Option D: [tex]\(x^4 + 20x^2 - 100\)[/tex].
