Answer :
To determine which expression is a prime polynomial, we need to check if each expression can be factored into simpler polynomials. A prime polynomial is one that cannot be factored further using integer or rational coefficients.
Let's examine each option:
A. [tex]\( 3x^2 + 18y \)[/tex]
- This expression can be factored by finding the greatest common factor of the terms, which is 3. Factoring out the 3 gives us:
[tex]\[ 3(x^2 + 6y) \][/tex]
- Since it can be factored, it is not a prime polynomial.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- This expression also can be factored. We can factor out an [tex]\( x \)[/tex] from all terms:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
- Since it can be factored, it is not a prime polynomial.
C. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes, which can be factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
- Here, [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex], giving:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
- Since it can be factored, it is not a prime polynomial.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression remains unfactorable over the integers or rationals. There are no common factors or special patterns (like the difference of squares, sum or difference of cubes) that allow it to be factored further.
- Since it cannot be factored, it is a prime polynomial.
Thus, the expression that is a prime polynomial is [tex]\( x^4 + 20x^2 - 100 \)[/tex] (Option D).
Let's examine each option:
A. [tex]\( 3x^2 + 18y \)[/tex]
- This expression can be factored by finding the greatest common factor of the terms, which is 3. Factoring out the 3 gives us:
[tex]\[ 3(x^2 + 6y) \][/tex]
- Since it can be factored, it is not a prime polynomial.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- This expression also can be factored. We can factor out an [tex]\( x \)[/tex] from all terms:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
- Since it can be factored, it is not a prime polynomial.
C. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes, which can be factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
- Here, [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex], giving:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
- Since it can be factored, it is not a prime polynomial.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression remains unfactorable over the integers or rationals. There are no common factors or special patterns (like the difference of squares, sum or difference of cubes) that allow it to be factored further.
- Since it cannot be factored, it is a prime polynomial.
Thus, the expression that is a prime polynomial is [tex]\( x^4 + 20x^2 - 100 \)[/tex] (Option D).
