Answer :
To determine which expression is a prime polynomial, we need to analyze each polynomial and see if it can be factored into simpler polynomials.
Let's go through each option:
### A. [tex]\( x^3 - 27y^6 \)[/tex]
This expression looks like a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
We can factor it using the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
So,
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
### 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 it can be factored, it is not a prime polynomial.
### C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
Let's try to factor this polynomial. First, we factor out the greatest common factor:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, it is not a prime polynomial.
### D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial can be factored as a quadratic in terms of [tex]\( x^2 \)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = (x^2 + 10)^2 - (10)^2 \][/tex]
This is a difference of squares:
[tex]\[ (x^2 + 10)^2 - 10^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) = (x^2)(x^2 + 20) \][/tex]
Since it can be factored, it is not a prime polynomial.
After analyzing each option, we can determine that none of these expressions are prime polynomials.
Thus, the correct answer is that none of the given options are prime polynomials.
Let's go through each option:
### A. [tex]\( x^3 - 27y^6 \)[/tex]
This expression looks like a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
We can factor it using the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
So,
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
### 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 it can be factored, it is not a prime polynomial.
### C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
Let's try to factor this polynomial. First, we factor out the greatest common factor:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, it is not a prime polynomial.
### D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial can be factored as a quadratic in terms of [tex]\( x^2 \)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = (x^2 + 10)^2 - (10)^2 \][/tex]
This is a difference of squares:
[tex]\[ (x^2 + 10)^2 - 10^2 = (x^2 + 10 - 10)(x^2 + 10 + 10) = (x^2)(x^2 + 20) \][/tex]
Since it can be factored, it is not a prime polynomial.
After analyzing each option, we can determine that none of these expressions are prime polynomials.
Thus, the correct answer is that none of the given options are prime polynomials.
