Answer :
To determine which expression is a prime polynomial, we need to check each option and see if it can be factored further. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with coefficients in the same field.
Let's analyze each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- To determine if this is a prime polynomial, we attempt to factor it. In this case, it cannot be factored into the product of two non-constant polynomials with real coefficients, so it is a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial can be factored out by taking a common factor or using other factoring techniques. It factors into simpler polynomials, which means it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
- This expression can be recognized as a difference of cubes. It can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex], so it is not a prime polynomial.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3. It can be written as [tex]\(3(x^2 + 6y)\)[/tex], so it is not a prime polynomial.
Therefore, the correct answer, where the expression is a prime polynomial, is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's analyze each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- To determine if this is a prime polynomial, we attempt to factor it. In this case, it cannot be factored into the product of two non-constant polynomials with real coefficients, so it is a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial can be factored out by taking a common factor or using other factoring techniques. It factors into simpler polynomials, which means it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
- This expression can be recognized as a difference of cubes. It can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex], so it is not a prime polynomial.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3. It can be written as [tex]\(3(x^2 + 6y)\)[/tex], so it is not a prime polynomial.
Therefore, the correct answer, where the expression is a prime polynomial, is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
