Answer :
To determine which expression is a prime polynomial from the options given, we'll analyze each polynomial:
1. Option A: [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- A prime polynomial is one that cannot be factored into polynomials of lower degrees with integer coefficients.
- This polynomial is a quadratic in terms of [tex]\( x^2 \)[/tex], written as [tex]\( (x^2)^2 + 20(x^2) - 100 \)[/tex].
- Factoring attempts either do not simplify the expression further with integer factors or confirm it's irreducible.
2. Option B: [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- This polynomial can often be factored by grouping or applying synthetic division, indicating it's not prime.
- In this case, it's reducible using these methods.
3. Option C: [tex]\( 3x^2 + 18y \)[/tex]
- Notice that this polynomial shares a common factor of 3.
- Factoring out yields [tex]\( 3(x^2 + 6y) \)[/tex], which shows it's not prime because it has been simplified.
4. Option D: [tex]\( x^3 - 27y^6 \)[/tex]
- This is a difference of cubes: [tex]\( x^3 - (3y^2)^3 \)[/tex].
- It can be factored using the difference of cubes formula: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex].
- Applying this formula confirms it can be reduced.
From the analysis, only Option A: [tex]\( x^4 + 20x^2 - 100 \)[/tex] remains un-factored with integer coefficients and is therefore a prime polynomial.
Thus, the correct answer is A.
1. Option A: [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- A prime polynomial is one that cannot be factored into polynomials of lower degrees with integer coefficients.
- This polynomial is a quadratic in terms of [tex]\( x^2 \)[/tex], written as [tex]\( (x^2)^2 + 20(x^2) - 100 \)[/tex].
- Factoring attempts either do not simplify the expression further with integer factors or confirm it's irreducible.
2. Option B: [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- This polynomial can often be factored by grouping or applying synthetic division, indicating it's not prime.
- In this case, it's reducible using these methods.
3. Option C: [tex]\( 3x^2 + 18y \)[/tex]
- Notice that this polynomial shares a common factor of 3.
- Factoring out yields [tex]\( 3(x^2 + 6y) \)[/tex], which shows it's not prime because it has been simplified.
4. Option D: [tex]\( x^3 - 27y^6 \)[/tex]
- This is a difference of cubes: [tex]\( x^3 - (3y^2)^3 \)[/tex].
- It can be factored using the difference of cubes formula: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex].
- Applying this formula confirms it can be reduced.
From the analysis, only Option A: [tex]\( x^4 + 20x^2 - 100 \)[/tex] remains un-factored with integer coefficients and is therefore a prime polynomial.
Thus, the correct answer is A.
