Answer :
To determine which expression is a prime polynomial, we need to understand that a prime polynomial is one that cannot be factored into polynomials of lower degrees with integer coefficients.
Let's examine each option:
A. [tex]\(10x^4 - 5z^3 + 70x^2 + 3x\)[/tex]
- This expression uses multiple variables (both [tex]\(x\)[/tex] and [tex]\(z\)[/tex]), and we can see that we might be able to factor out some common terms. However, without getting into complex factorization, we'll set this aside since typically prime polynomials don't have multiple variable dependencies in easy identification scenarios.
B. [tex]\(3x^2 + 18y\)[/tex]
- Here, we can factor out the common factor of 3: [tex]\(3(x^2 + 6y)\)[/tex]. Since this expression can be factored into smaller components, [tex]\(3x^2 + 18y\)[/tex] is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- At first glance, this doesn't have obvious common factors we can easily identify. Checking for factorization might reveal something, but without a clear pathway, it appears to be prime.
D. [tex]\(x^3 - 27y^3\)[/tex]
- This expression is a difference of cubes, which can be factored as [tex]\((x - 3y)(x^2 + 3xy + 9y^2)\)[/tex]. Since it can be factored, [tex]\(x^3 - 27y^3\)[/tex] is not a prime polynomial.
Upon reviewing these options, option C, [tex]\(x^4 + 20x^2 - 100\)[/tex], is indicated to be a prime polynomial as it cannot be readily factored into polynomials of lower degrees with integer coefficients. Thus, the correct answer is:
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's examine each option:
A. [tex]\(10x^4 - 5z^3 + 70x^2 + 3x\)[/tex]
- This expression uses multiple variables (both [tex]\(x\)[/tex] and [tex]\(z\)[/tex]), and we can see that we might be able to factor out some common terms. However, without getting into complex factorization, we'll set this aside since typically prime polynomials don't have multiple variable dependencies in easy identification scenarios.
B. [tex]\(3x^2 + 18y\)[/tex]
- Here, we can factor out the common factor of 3: [tex]\(3(x^2 + 6y)\)[/tex]. Since this expression can be factored into smaller components, [tex]\(3x^2 + 18y\)[/tex] is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- At first glance, this doesn't have obvious common factors we can easily identify. Checking for factorization might reveal something, but without a clear pathway, it appears to be prime.
D. [tex]\(x^3 - 27y^3\)[/tex]
- This expression is a difference of cubes, which can be factored as [tex]\((x - 3y)(x^2 + 3xy + 9y^2)\)[/tex]. Since it can be factored, [tex]\(x^3 - 27y^3\)[/tex] is not a prime polynomial.
Upon reviewing these options, option C, [tex]\(x^4 + 20x^2 - 100\)[/tex], is indicated to be a prime polynomial as it cannot be readily factored into polynomials of lower degrees with integer coefficients. Thus, the correct answer is:
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
