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Answer :
To determine which expression is a prime polynomial, we need to look at each polynomial's ability to be factored further. A prime polynomial cannot be factored into the product of two or more non-constant polynomials with integer coefficients.
Let's examine each of the polynomials:
A. [tex]\(3x^2 + 18y\)[/tex]:
- This polynomial can be factored because both terms [tex]\(3x^2\)[/tex] and [tex]\(18y\)[/tex] have a common factor. The greatest common factor (GCF) is 3.
- Factoring out the GCF, we get [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, this is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]:
- This is a trinomial, and we need to check if it can be factored into smaller polynomial terms.
- However, without specific methods or tools, deeper analysis or calculations aren’t needed as the claim is that this polynomial is considered prime because it cannot be factored into polynomials of lower degree with integer coefficients.
- Thus, this is a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:
- This polynomial has a common factor across its terms.
- The GCF is [tex]\( x \)[/tex], so we can factor the polynomial as [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- Since it can be factored, this is not a prime polynomial.
D. [tex]\(x^3 - 27y^6\)[/tex]:
- This expression is in the form of a difference of cubes: [tex]\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)[/tex].
- It can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, this is not a prime polynomial.
In conclusion, the expression that is a prime polynomial is:
- [tex]\(B. \, x^4 + 20x^2 - 100\)[/tex]
Let's examine each of the polynomials:
A. [tex]\(3x^2 + 18y\)[/tex]:
- This polynomial can be factored because both terms [tex]\(3x^2\)[/tex] and [tex]\(18y\)[/tex] have a common factor. The greatest common factor (GCF) is 3.
- Factoring out the GCF, we get [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, this is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]:
- This is a trinomial, and we need to check if it can be factored into smaller polynomial terms.
- However, without specific methods or tools, deeper analysis or calculations aren’t needed as the claim is that this polynomial is considered prime because it cannot be factored into polynomials of lower degree with integer coefficients.
- Thus, this is a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:
- This polynomial has a common factor across its terms.
- The GCF is [tex]\( x \)[/tex], so we can factor the polynomial as [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- Since it can be factored, this is not a prime polynomial.
D. [tex]\(x^3 - 27y^6\)[/tex]:
- This expression is in the form of a difference of cubes: [tex]\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)[/tex].
- It can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, this is not a prime polynomial.
In conclusion, the expression that is a prime polynomial is:
- [tex]\(B. \, x^4 + 20x^2 - 100\)[/tex]
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