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Answer :
To determine which expression is a prime polynomial, we need to figure out if each expression can be factored further. A prime polynomial cannot be factored into polynomials of lower degree with integer or rational coefficients.
Let's examine each expression:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor an [tex]\(x\)[/tex] out of each term: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. The polynomial inside the parenthesis isn't easily factorable into polynomials of lower degree, but since we factored out [tex]\(x\)[/tex], this isn't a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This is a difference of cubes because [tex]\(27y^6\)[/tex] can be written as [tex]\((3y^2)^3\)[/tex]. It factors as [tex]\(x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
C. [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a 3 from each term: [tex]\(3(x^2 + 6y)\)[/tex]. The polynomial inside the parenthesis isn't further factorable with integer coefficients, but since we factored out 3, this isn't a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- We can use substitution by letting [tex]\(u = x^2\)[/tex], transforming the expression into a quadratic: [tex]\(u^2 + 20u - 100\)[/tex]. This quadratic can potentially be factored further, but the substitution shows it's not immediately prime.
Given these examinations, none of these expressions are prime polynomials in terms of being unfactorable at a basic level. The expressions provided all have some factorization, indicating none of them fits the strictest definition of a prime polynomial using integer or rational coefficients.
Let's examine each expression:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor an [tex]\(x\)[/tex] out of each term: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. The polynomial inside the parenthesis isn't easily factorable into polynomials of lower degree, but since we factored out [tex]\(x\)[/tex], this isn't a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This is a difference of cubes because [tex]\(27y^6\)[/tex] can be written as [tex]\((3y^2)^3\)[/tex]. It factors as [tex]\(x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
C. [tex]\(3x^2 + 18y\)[/tex]
- We can factor out a 3 from each term: [tex]\(3(x^2 + 6y)\)[/tex]. The polynomial inside the parenthesis isn't further factorable with integer coefficients, but since we factored out 3, this isn't a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- We can use substitution by letting [tex]\(u = x^2\)[/tex], transforming the expression into a quadratic: [tex]\(u^2 + 20u - 100\)[/tex]. This quadratic can potentially be factored further, but the substitution shows it's not immediately prime.
Given these examinations, none of these expressions are prime polynomials in terms of being unfactorable at a basic level. The expressions provided all have some factorization, indicating none of them fits the strictest definition of a prime polynomial using integer or rational coefficients.
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