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Answer :
To determine which expression is a prime polynomial, let’s understand what a prime polynomial is. A prime polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials over the integers.
Let's go through each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial is already given in a form where it cannot be further factored over the integers, which means it does not have any factors other than itself and 1.
B. [tex]\( x^3 - 27y^6 \)[/tex]
This expression can be factored as [tex]\( (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \)[/tex]. Since it can be factored into products of polynomials with integer coefficients, it is not a prime polynomial.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This polynomial can be factored as [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex]. Since it can be broken down further, it is not a prime polynomial.
D. [tex]\( 3x^2 + 18y \)[/tex]
This can be factored as [tex]\( 3(x^2 + 6y) \)[/tex]. Because it can be factored, it is not a prime polynomial.
After evaluating each of the four options, we see that the polynomial in option A, [tex]\( x^4 + 20x^2 - 100 \)[/tex], is a prime polynomial because it cannot be factored into smaller polynomials with integer coefficients. Therefore, the correct answer is A.
Let's go through each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial is already given in a form where it cannot be further factored over the integers, which means it does not have any factors other than itself and 1.
B. [tex]\( x^3 - 27y^6 \)[/tex]
This expression can be factored as [tex]\( (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \)[/tex]. Since it can be factored into products of polynomials with integer coefficients, it is not a prime polynomial.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
This polynomial can be factored as [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex]. Since it can be broken down further, it is not a prime polynomial.
D. [tex]\( 3x^2 + 18y \)[/tex]
This can be factored as [tex]\( 3(x^2 + 6y) \)[/tex]. Because it can be factored, it is not a prime polynomial.
After evaluating each of the four options, we see that the polynomial in option A, [tex]\( x^4 + 20x^2 - 100 \)[/tex], is a prime polynomial because it cannot be factored into smaller polynomials with integer coefficients. Therefore, the correct answer is A.
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