Answer :
Sure, let's determine which of these polynomials is a prime polynomial. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with coefficients in the same field or ring. Here is what you need to know:
1. Option A: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Look for common factors in the terms. Here, we can factor out the greatest common factor, which is [tex]\(x\)[/tex].
- This expression can be rewritten as [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since it can be factored further, this is not a prime polynomial.
2. Option B: [tex]\(z^3 - 27y^8\)[/tex]
- This expression is reminiscent of a difference of cubes, since [tex]\(27y^8\)[/tex] can be written as [tex]\((3y^{8/3})^3\)[/tex].
- The expression can be factored into [tex]\((z - 3y^{8/3})(z^2 + 3y^{8/3}z + (3y^{8/3})^2)\)[/tex].
- As it can be factored, it is not a prime polynomial.
3. Option C: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- Check if it can be factored by trial or by using factoring techniques.
- This polynomial can be transformed into something that might resemble a factorable quadratic by substituting [tex]\(u = x^2\)[/tex], giving us [tex]\(u^2 + 20u - 100\)[/tex], which can be factored using techniques for quadratics.
- Factoring does show that this expression can be further simplified, meaning it's also not a prime polynomial.
4. Option D: [tex]\(3x^2 + 18y\)[/tex]
- Look for a common factor. We can factor out a 3, getting [tex]\(3(x^2 + 6y)\)[/tex].
- Since the expression can be simplified further, it is not a prime polynomial.
Based on these considerations, none of these polynomials is prime as they all can be factored. However, if we're going strictly by the results provided, it seems that Options B and C returned as prime, which may not have reflected my step-by-step breakdown, since they all had the potential to be factored. Therefore, none logically should be prime without specifying unique or erroneous conditions.
1. Option A: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Look for common factors in the terms. Here, we can factor out the greatest common factor, which is [tex]\(x\)[/tex].
- This expression can be rewritten as [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since it can be factored further, this is not a prime polynomial.
2. Option B: [tex]\(z^3 - 27y^8\)[/tex]
- This expression is reminiscent of a difference of cubes, since [tex]\(27y^8\)[/tex] can be written as [tex]\((3y^{8/3})^3\)[/tex].
- The expression can be factored into [tex]\((z - 3y^{8/3})(z^2 + 3y^{8/3}z + (3y^{8/3})^2)\)[/tex].
- As it can be factored, it is not a prime polynomial.
3. Option C: [tex]\(x^4 + 20x^2 - 100\)[/tex]
- Check if it can be factored by trial or by using factoring techniques.
- This polynomial can be transformed into something that might resemble a factorable quadratic by substituting [tex]\(u = x^2\)[/tex], giving us [tex]\(u^2 + 20u - 100\)[/tex], which can be factored using techniques for quadratics.
- Factoring does show that this expression can be further simplified, meaning it's also not a prime polynomial.
4. Option D: [tex]\(3x^2 + 18y\)[/tex]
- Look for a common factor. We can factor out a 3, getting [tex]\(3(x^2 + 6y)\)[/tex].
- Since the expression can be simplified further, it is not a prime polynomial.
Based on these considerations, none of these polynomials is prime as they all can be factored. However, if we're going strictly by the results provided, it seems that Options B and C returned as prime, which may not have reflected my step-by-step breakdown, since they all had the potential to be factored. Therefore, none logically should be prime without specifying unique or erroneous conditions.
