Answer :
Sure, let's determine which expression is a prime polynomial by analyzing each of the options provided.
### Understanding Prime Polynomials
A prime polynomial (or irreducible polynomial) is a polynomial that cannot be factored into the product of two non-constant polynomials with coefficients in the same field.
### Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
First, we should check if this polynomial can be factored:
1. [tex]\(x^4 + 20x^2 - 100\)[/tex]
The polynomial can be factored as follows:
[tex]\[ x^4 + 20x^2 - 100 = x^2(x^2 + 20) - 100 = (x^2 + 10)^2 - 10^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) = (x^2)(x^2 + 20) - 100 \][/tex]
However, it does not simplify this way, but consider the perfect square completion and differences. The polynomial may be reducible in a complex field. This analysis indicates that this polynomial is reducible.
### Option B: [tex]\(x^3 - 27y^6\)[/tex]
This polynomial can be expressed in terms of a difference of cubes:
1. Recognize the difference of cubes:
[tex]\[ x^3 - 27y^6 = x^3 - (3y^2)^3 \][/tex]
The difference of cubes formula is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Applying this formula:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
This shows that [tex]\(x^3 - 27y^6\)[/tex] is not a prime polynomial.
### Option C: [tex]\(3x^2 + 18y\)[/tex]
We notice a common factor here:
1. Factor out the greatest common divisor:
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
This factorization shows that [tex]\(3x^2 + 18y\)[/tex] is not a prime polynomial.
### Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
We can attempt to factor this polynomial by searching for common factors or possible groupings:
1. Check for a common factor:
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
This factorization shows that the polynomial contains a factor of [tex]\(x\)[/tex], indicating that it is not prime.
### Conclusion
Among the given options, the polynomial expression that cannot be factored further and remains irreducible is:
[tex]\[ x^4 + 20x^2 - 100 \][/tex]
So the correct answer is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
### Understanding Prime Polynomials
A prime polynomial (or irreducible polynomial) is a polynomial that cannot be factored into the product of two non-constant polynomials with coefficients in the same field.
### Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
First, we should check if this polynomial can be factored:
1. [tex]\(x^4 + 20x^2 - 100\)[/tex]
The polynomial can be factored as follows:
[tex]\[ x^4 + 20x^2 - 100 = x^2(x^2 + 20) - 100 = (x^2 + 10)^2 - 10^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) = (x^2)(x^2 + 20) - 100 \][/tex]
However, it does not simplify this way, but consider the perfect square completion and differences. The polynomial may be reducible in a complex field. This analysis indicates that this polynomial is reducible.
### Option B: [tex]\(x^3 - 27y^6\)[/tex]
This polynomial can be expressed in terms of a difference of cubes:
1. Recognize the difference of cubes:
[tex]\[ x^3 - 27y^6 = x^3 - (3y^2)^3 \][/tex]
The difference of cubes formula is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Applying this formula:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
This shows that [tex]\(x^3 - 27y^6\)[/tex] is not a prime polynomial.
### Option C: [tex]\(3x^2 + 18y\)[/tex]
We notice a common factor here:
1. Factor out the greatest common divisor:
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
This factorization shows that [tex]\(3x^2 + 18y\)[/tex] is not a prime polynomial.
### Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
We can attempt to factor this polynomial by searching for common factors or possible groupings:
1. Check for a common factor:
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
This factorization shows that the polynomial contains a factor of [tex]\(x\)[/tex], indicating that it is not prime.
### Conclusion
Among the given options, the polynomial expression that cannot be factored further and remains irreducible is:
[tex]\[ x^4 + 20x^2 - 100 \][/tex]
So the correct answer is:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
