Answer :
To determine which expression is a prime polynomial from the given list, we need to check if any of the polynomials cannot be factored into polynomials of lower degree with integer coefficients. A prime polynomial cannot be factored further.
Let's review each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial has more than just one term and might not be easily factorable. However, to confirm if it's truly prime, we'd try possible factorizations involving numbers or variable expressions. This one might simplify by considering possible factors, but it seems complex enough to suspect it could be prime.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression resembles a difference of cubes. Remember that [tex]\(a^3 - b^3\)[/tex] can be factored into [tex]\((a - b)(a^2 + ab + b^2)\)[/tex]. Using that rule, this polynomial can be factored, and thus is not prime.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression has a common factor in its terms. Let's factor out the greatest common factor, which is [tex]\(x\)[/tex], giving [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. This indicates that it's not prime since it can be factored.
D. [tex]\(3x^2 + 18y\)[/tex]
- The terms share a common factor of 3. Factoring out the 3 gives [tex]\(3(x^2 + 6y)\)[/tex]. Since we have successfully factored the polynomial, it is not prime.
In summary, analyzing all of the options, each polynomial can be factored in some way, meaning they are not prime polynomials. Therefore, none of the given expressions is a prime polynomial.
Let's review each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial has more than just one term and might not be easily factorable. However, to confirm if it's truly prime, we'd try possible factorizations involving numbers or variable expressions. This one might simplify by considering possible factors, but it seems complex enough to suspect it could be prime.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression resembles a difference of cubes. Remember that [tex]\(a^3 - b^3\)[/tex] can be factored into [tex]\((a - b)(a^2 + ab + b^2)\)[/tex]. Using that rule, this polynomial can be factored, and thus is not prime.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression has a common factor in its terms. Let's factor out the greatest common factor, which is [tex]\(x\)[/tex], giving [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. This indicates that it's not prime since it can be factored.
D. [tex]\(3x^2 + 18y\)[/tex]
- The terms share a common factor of 3. Factoring out the 3 gives [tex]\(3(x^2 + 6y)\)[/tex]. Since we have successfully factored the polynomial, it is not prime.
In summary, analyzing all of the options, each polynomial can be factored in some way, meaning they are not prime polynomials. Therefore, none of the given expressions is a prime polynomial.
