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Answer :
- Check if each polynomial can be factored.
- A. $x^4+20 x^2-100$ can be factored.
- B. $10 x^4-5 x^3+70 x^2+3 x$ can be factored.
- C. $x^3-27 y^6$ can be factored.
- D. $3 x^2+18 y$ is a prime polynomial. The final answer is $\boxed{D}$.
### Explanation
1. Understanding Prime Polynomials
We are given four polynomial expressions and asked to identify which one is a prime polynomial. A prime polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials of lower degree. Let's analyze each option:
2. Analyzing Option A
A. $x^4+20 x^2-100$. This is a quadratic in $x^2$. Let $y = x^2$. Then we have $y^2 + 20y - 100$. The discriminant is $b^2 - 4ac = 20^2 - 4(1)(-100) = 400 + 400 = 800$. Since the discriminant is positive but not a perfect square, the roots are real but irrational. Thus, it can be factored as $(x^2 - r_1)(x^2 - r_2)$, where $r_1$ and $r_2$ are the roots. So, it is not prime.
3. Analyzing Option B
B. $10 x^4-5 x^3+70 x^2+3 x = x(10x^3 - 5x^2 + 70x + 3)$. Since it can be factored as $x$ times another polynomial, it is not prime.
4. Analyzing Option C
C. $x^3-27 y^6 = x^3 - (3y^2)^3$. This is a difference of cubes, so it can be factored as $(x-3y^2)(x^2 + 3xy^2 + 9y^4)$. Thus, it is not prime.
5. Analyzing Option D
D. $3 x^2+18 y = 3(x^2 + 6y)$. We can factor out a 3. The expression $x^2 + 6y$ cannot be factored further into non-constant polynomials. Therefore, $3 x^2+18 y$ is a prime polynomial.
6. Conclusion
Based on the analysis above, the prime polynomial is $3x^2 + 18y$.
### Examples
Prime polynomials are analogous to prime numbers in that they cannot be factored into simpler polynomials, just as prime numbers cannot be factored into smaller integers. Identifying prime polynomials is essential in various areas of mathematics, including abstract algebra and cryptography. For instance, in coding theory, prime polynomials are used to construct error-correcting codes, which are crucial for reliable data transmission. Understanding prime polynomials helps in designing efficient and secure communication systems.
- A. $x^4+20 x^2-100$ can be factored.
- B. $10 x^4-5 x^3+70 x^2+3 x$ can be factored.
- C. $x^3-27 y^6$ can be factored.
- D. $3 x^2+18 y$ is a prime polynomial. The final answer is $\boxed{D}$.
### Explanation
1. Understanding Prime Polynomials
We are given four polynomial expressions and asked to identify which one is a prime polynomial. A prime polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials of lower degree. Let's analyze each option:
2. Analyzing Option A
A. $x^4+20 x^2-100$. This is a quadratic in $x^2$. Let $y = x^2$. Then we have $y^2 + 20y - 100$. The discriminant is $b^2 - 4ac = 20^2 - 4(1)(-100) = 400 + 400 = 800$. Since the discriminant is positive but not a perfect square, the roots are real but irrational. Thus, it can be factored as $(x^2 - r_1)(x^2 - r_2)$, where $r_1$ and $r_2$ are the roots. So, it is not prime.
3. Analyzing Option B
B. $10 x^4-5 x^3+70 x^2+3 x = x(10x^3 - 5x^2 + 70x + 3)$. Since it can be factored as $x$ times another polynomial, it is not prime.
4. Analyzing Option C
C. $x^3-27 y^6 = x^3 - (3y^2)^3$. This is a difference of cubes, so it can be factored as $(x-3y^2)(x^2 + 3xy^2 + 9y^4)$. Thus, it is not prime.
5. Analyzing Option D
D. $3 x^2+18 y = 3(x^2 + 6y)$. We can factor out a 3. The expression $x^2 + 6y$ cannot be factored further into non-constant polynomials. Therefore, $3 x^2+18 y$ is a prime polynomial.
6. Conclusion
Based on the analysis above, the prime polynomial is $3x^2 + 18y$.
### Examples
Prime polynomials are analogous to prime numbers in that they cannot be factored into simpler polynomials, just as prime numbers cannot be factored into smaller integers. Identifying prime polynomials is essential in various areas of mathematics, including abstract algebra and cryptography. For instance, in coding theory, prime polynomials are used to construct error-correcting codes, which are crucial for reliable data transmission. Understanding prime polynomials helps in designing efficient and secure communication systems.
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