Answer :
Sure, I’d be happy to help you with that! To determine which expression is a prime polynomial, let’s check if each polynomial can be factored further. A prime polynomial is one that cannot be factored into polynomials of lower degrees with rational coefficients.
### Analyzing Each Polynomial:
1. Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- Rewrite it as: [tex]\(x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100\)[/tex].
- Let [tex]\(u = x^2\)[/tex]. Then it becomes a quadratic in [tex]\(u\)[/tex]: [tex]\(u^2 + 20u - 100\)[/tex].
- Solve the quadratic equation [tex]\(u^2 + 20u - 100 = 0\)[/tex].
- Discriminant [tex]\(\Delta = 20^2 - 4 \cdot 1 \cdot (-100) = 400 + 400 = 800\)[/tex].
- The discriminant is positive, indicating the quadratic can be factored. Therefore, it is not prime.
2. Option B: [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by taking out the greatest common factor: [tex]\(3(x^2 + 6y)\)[/tex].
- Since [tex]\(3\)[/tex] is a common factor, it is not prime.
3. Option C: [tex]\(x^3 - 27y^6\)[/tex]
- Recognize it as a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
- Use the formula for the difference of cubes: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
- Applying the formula gives: [tex]\(x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, it is not prime.
4. Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Factor out the greatest common factor which is [tex]\(x\)[/tex]: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since we can factor out [tex]\(x\)[/tex], it is not prime.
### Conclusion:
After analyzing each option:
- All the given polynomials can be factored and are not prime.
However, considering typical problems, let's review Option A one more time:
- If we additionally check the factorization extensively and their roots do not provide rational solutions that facilitate factoring [tex]\(x^4+20x^2-100 \equiv 0\)[/tex], then possibly this polynomial might be considered as irreducible overall in typical exam settings.
Thus,:
None of the polynomials provided in the options remain strictly prime under basic factor checking often given in problem contexts.
So rechecking reveals provided options tend to non-prime on available factor insight, general contexts for prime checks need to be considered deeply beyond basic factorizing.
Recommendation: Cross-verifications or deeper checks with polynomial standard bound books may sharpen prime-identification further.
### Analyzing Each Polynomial:
1. Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- Rewrite it as: [tex]\(x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100\)[/tex].
- Let [tex]\(u = x^2\)[/tex]. Then it becomes a quadratic in [tex]\(u\)[/tex]: [tex]\(u^2 + 20u - 100\)[/tex].
- Solve the quadratic equation [tex]\(u^2 + 20u - 100 = 0\)[/tex].
- Discriminant [tex]\(\Delta = 20^2 - 4 \cdot 1 \cdot (-100) = 400 + 400 = 800\)[/tex].
- The discriminant is positive, indicating the quadratic can be factored. Therefore, it is not prime.
2. Option B: [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by taking out the greatest common factor: [tex]\(3(x^2 + 6y)\)[/tex].
- Since [tex]\(3\)[/tex] is a common factor, it is not prime.
3. Option C: [tex]\(x^3 - 27y^6\)[/tex]
- Recognize it as a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
- Use the formula for the difference of cubes: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex].
- Applying the formula gives: [tex]\(x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, it is not prime.
4. Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Factor out the greatest common factor which is [tex]\(x\)[/tex]: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since we can factor out [tex]\(x\)[/tex], it is not prime.
### Conclusion:
After analyzing each option:
- All the given polynomials can be factored and are not prime.
However, considering typical problems, let's review Option A one more time:
- If we additionally check the factorization extensively and their roots do not provide rational solutions that facilitate factoring [tex]\(x^4+20x^2-100 \equiv 0\)[/tex], then possibly this polynomial might be considered as irreducible overall in typical exam settings.
Thus,:
None of the polynomials provided in the options remain strictly prime under basic factor checking often given in problem contexts.
So rechecking reveals provided options tend to non-prime on available factor insight, general contexts for prime checks need to be considered deeply beyond basic factorizing.
Recommendation: Cross-verifications or deeper checks with polynomial standard bound books may sharpen prime-identification further.
