Answer :
To find out which expression is a prime polynomial, let's analyze each option one by one.
A prime polynomial is a polynomial that cannot be factored into the product of two or more non-constant polynomials with coefficients in the same field (for example, real numbers).
Let's look at each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This expression is not prime because it can be factored. We notice that it has a quadratic form in terms of [tex]\( x^2 \)[/tex], which means we can set [tex]\( u = x^2 \)[/tex] and rewrite it as [tex]\( u^2 + 20u - 100 \)[/tex], which can be factored further.
B. [tex]\( 3x^2 + 18y \)[/tex]
This polynomial can also be factored because we can take out the greatest common factor, which is 3. The expression becomes [tex]\( 3(x^2 + 6y) \)[/tex], showing it is not prime.
C. [tex]\( x^3 - 27y^6 \)[/tex]
This expression matches the form of a difference of cubes: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]. Here, [tex]\( a = x \)[/tex] and [tex]\( b = (3y^2) \)[/tex]. So, it can be factored as [tex]\( (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \)[/tex], which means it is not prime.
D. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
We can factor out the greatest common factor, which is [tex]\( x \)[/tex], so the expression becomes [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex]. This indicates it is not prime.
Since all four options can be factored further, none of them are prime polynomials. Thus, according to the information, each expression can be factored, and there is no prime polynomial among the given options.
A prime polynomial is a polynomial that cannot be factored into the product of two or more non-constant polynomials with coefficients in the same field (for example, real numbers).
Let's look at each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This expression is not prime because it can be factored. We notice that it has a quadratic form in terms of [tex]\( x^2 \)[/tex], which means we can set [tex]\( u = x^2 \)[/tex] and rewrite it as [tex]\( u^2 + 20u - 100 \)[/tex], which can be factored further.
B. [tex]\( 3x^2 + 18y \)[/tex]
This polynomial can also be factored because we can take out the greatest common factor, which is 3. The expression becomes [tex]\( 3(x^2 + 6y) \)[/tex], showing it is not prime.
C. [tex]\( x^3 - 27y^6 \)[/tex]
This expression matches the form of a difference of cubes: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]. Here, [tex]\( a = x \)[/tex] and [tex]\( b = (3y^2) \)[/tex]. So, it can be factored as [tex]\( (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \)[/tex], which means it is not prime.
D. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
We can factor out the greatest common factor, which is [tex]\( x \)[/tex], so the expression becomes [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex]. This indicates it is not prime.
Since all four options can be factored further, none of them are prime polynomials. Thus, according to the information, each expression can be factored, and there is no prime polynomial among the given options.
