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Answer :
- Check if each polynomial can be factored.
- A. $x^4+20 x^2-100$ can be factored, so it is not prime.
- B. $x^3-27 y^6$ can be factored, so it is not prime.
- C. $3 x^2+18 y$ cannot be factored into non-constant polynomials, so it is prime.
- D. $10 x^4-5 x^3+70 x^2+3 x$ can be factored, so it is not prime.
- The prime polynomial is $\boxed{C}$.
### 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.
2. Analyzing Each Option
Let's analyze each option:
A. $x^4+20 x^2-100$. From the tool, we see that this polynomial does not factor nicely over the integers, remaining as $x^4 + 20x^2 - 100$. However, this doesn't immediately mean it's prime. We need to check if it can be factored over real numbers. Let $y = x^2$. Then we have $y^2 + 20y - 100$. Using the quadratic formula, $y = \frac{-20 \pm \sqrt{20^2 - 4(1)(-100)}}{2} = \frac{-20 \pm \sqrt{800}}{2} = -10 \pm 10\sqrt{2}$. Thus, $x^2 = -10 \pm 10\sqrt{2}$, which means $x = \pm \sqrt{-10 \pm 10\sqrt{2}}$. Since it can be factored into non-constant polynomials, A is not prime.
B. $x^3-27 y^6$. From the tool, we see that $x^3-27 y^6 = (x-3y^2)(x^2 + 3xy^2 + 9y^4)$. Since it can be factored into non-constant polynomials, B is not prime.
C. $3 x^2+18 y = 3(x^2 + 6y)$. This expression can be factored as $3(x^2 + 6y)$. The term $x^2 + 6y$ cannot be factored further into non-constant polynomials. Thus, C is a prime polynomial.
D. $10 x^4-5 x^3+70 x^2+3 x = x(10x^3 - 5x^2 + 70x + 3)$. Since it can be factored into non-constant polynomials, D is not prime.
3. Conclusion
Therefore, the prime polynomial is C. $3x^2 + 18y$.
### Examples
Prime polynomials are similar to prime numbers in that they cannot be factored into simpler expressions. In cryptography, prime numbers are used to create secure codes. Similarly, in coding theory, prime polynomials can be used to construct error-correcting codes, which help ensure that data is transmitted accurately. Understanding prime polynomials helps in designing robust and reliable communication systems.
- A. $x^4+20 x^2-100$ can be factored, so it is not prime.
- B. $x^3-27 y^6$ can be factored, so it is not prime.
- C. $3 x^2+18 y$ cannot be factored into non-constant polynomials, so it is prime.
- D. $10 x^4-5 x^3+70 x^2+3 x$ can be factored, so it is not prime.
- The prime polynomial is $\boxed{C}$.
### 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.
2. Analyzing Each Option
Let's analyze each option:
A. $x^4+20 x^2-100$. From the tool, we see that this polynomial does not factor nicely over the integers, remaining as $x^4 + 20x^2 - 100$. However, this doesn't immediately mean it's prime. We need to check if it can be factored over real numbers. Let $y = x^2$. Then we have $y^2 + 20y - 100$. Using the quadratic formula, $y = \frac{-20 \pm \sqrt{20^2 - 4(1)(-100)}}{2} = \frac{-20 \pm \sqrt{800}}{2} = -10 \pm 10\sqrt{2}$. Thus, $x^2 = -10 \pm 10\sqrt{2}$, which means $x = \pm \sqrt{-10 \pm 10\sqrt{2}}$. Since it can be factored into non-constant polynomials, A is not prime.
B. $x^3-27 y^6$. From the tool, we see that $x^3-27 y^6 = (x-3y^2)(x^2 + 3xy^2 + 9y^4)$. Since it can be factored into non-constant polynomials, B is not prime.
C. $3 x^2+18 y = 3(x^2 + 6y)$. This expression can be factored as $3(x^2 + 6y)$. The term $x^2 + 6y$ cannot be factored further into non-constant polynomials. Thus, C is a prime polynomial.
D. $10 x^4-5 x^3+70 x^2+3 x = x(10x^3 - 5x^2 + 70x + 3)$. Since it can be factored into non-constant polynomials, D is not prime.
3. Conclusion
Therefore, the prime polynomial is C. $3x^2 + 18y$.
### Examples
Prime polynomials are similar to prime numbers in that they cannot be factored into simpler expressions. In cryptography, prime numbers are used to create secure codes. Similarly, in coding theory, prime polynomials can be used to construct error-correcting codes, which help ensure that data is transmitted accurately. Understanding prime polynomials helps in designing robust and reliable communication systems.
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