Answer :
- Option A factors as $3(x^2 + 6y)$.
- Option B factors as $x(10x^3 - 5x^2 + 70x + 3)$.
- Option C, $x^4 + 20x^2 - 100$, factors into $(x^2 + 10 - 10\sqrt{2})(x^2 + 10 + 10\sqrt{2})$. If we consider only integer coefficients, this polynomial is prime.
- Option D factors as $(x - 3y^2)(x^2 + 3xy^2 + 9y^4)$.
- Therefore, the prime polynomial is $\boxed{x^4+20 x^2-100}$.
### Explanation
1. Understanding Prime Polynomials
A prime polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. We need to check each option to see if it can be factored.
2. Analyzing Option A
A. $3x^2 + 18y = 3(x^2 + 6y)$. This can be factored, so it is not a prime polynomial.
3. Analyzing Option B
B. $10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)$. This can be factored, so it is not a prime polynomial.
4. Analyzing Option C
C. $x^4 + 20x^2 - 100$. Let $u = x^2$. Then we have $u^2 + 20u - 100$. The discriminant is $20^2 - 4(1)(-100) = 400 + 400 = 800$. Since the discriminant is not a perfect square, the quadratic does not factor with integer coefficients. However, this does not mean the original polynomial is prime. We can complete the square: $x^4 + 20x^2 + 100 - 200 = (x^2 + 10)^2 - (10\sqrt{2})^2 = (x^2 + 10 - 10\sqrt{2})(x^2 + 10 + 10\sqrt{2})$. Since it can be factored, it is not a prime polynomial.
5. Analyzing Option D
D. $x^3 - 27y^6 = x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)$. This can be factored, so it is not a prime polynomial.
6. Final Answer and Conclusion
Therefore, none of the given polynomials are prime. However, since we must select the correct answer, and the question implies that one of them *is* prime, let's re-examine option C. We found that $x^4 + 20x^2 - 100 = (x^2 + 10 - 10\sqrt{2})(x^2 + 10 + 10\sqrt{2})$. This factorization involves irrational coefficients. If we restrict ourselves to polynomials with integer (or rational) coefficients, then $x^4 + 20x^2 - 100$ is indeed a prime polynomial.
### Examples
Prime polynomials are analogous to prime numbers in that they cannot be factored into simpler polynomials. This concept is crucial in cryptography, where prime polynomials are used to construct finite fields for secure data encryption. Understanding prime polynomials helps in designing robust encryption algorithms, ensuring that sensitive information remains protected during transmission and storage.
- Option B factors as $x(10x^3 - 5x^2 + 70x + 3)$.
- Option C, $x^4 + 20x^2 - 100$, factors into $(x^2 + 10 - 10\sqrt{2})(x^2 + 10 + 10\sqrt{2})$. If we consider only integer coefficients, this polynomial is prime.
- Option D factors as $(x - 3y^2)(x^2 + 3xy^2 + 9y^4)$.
- Therefore, the prime polynomial is $\boxed{x^4+20 x^2-100}$.
### Explanation
1. Understanding Prime Polynomials
A prime polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. We need to check each option to see if it can be factored.
2. Analyzing Option A
A. $3x^2 + 18y = 3(x^2 + 6y)$. This can be factored, so it is not a prime polynomial.
3. Analyzing Option B
B. $10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)$. This can be factored, so it is not a prime polynomial.
4. Analyzing Option C
C. $x^4 + 20x^2 - 100$. Let $u = x^2$. Then we have $u^2 + 20u - 100$. The discriminant is $20^2 - 4(1)(-100) = 400 + 400 = 800$. Since the discriminant is not a perfect square, the quadratic does not factor with integer coefficients. However, this does not mean the original polynomial is prime. We can complete the square: $x^4 + 20x^2 + 100 - 200 = (x^2 + 10)^2 - (10\sqrt{2})^2 = (x^2 + 10 - 10\sqrt{2})(x^2 + 10 + 10\sqrt{2})$. Since it can be factored, it is not a prime polynomial.
5. Analyzing Option D
D. $x^3 - 27y^6 = x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)$. This can be factored, so it is not a prime polynomial.
6. Final Answer and Conclusion
Therefore, none of the given polynomials are prime. However, since we must select the correct answer, and the question implies that one of them *is* prime, let's re-examine option C. We found that $x^4 + 20x^2 - 100 = (x^2 + 10 - 10\sqrt{2})(x^2 + 10 + 10\sqrt{2})$. This factorization involves irrational coefficients. If we restrict ourselves to polynomials with integer (or rational) coefficients, then $x^4 + 20x^2 - 100$ is indeed a prime polynomial.
### Examples
Prime polynomials are analogous to prime numbers in that they cannot be factored into simpler polynomials. This concept is crucial in cryptography, where prime polynomials are used to construct finite fields for secure data encryption. Understanding prime polynomials helps in designing robust encryption algorithms, ensuring that sensitive information remains protected during transmission and storage.
