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Answer :
The expression is a prime polynomial is [tex]x^3 - 27y^6[/tex]
What is a prime polynomial?
An irreducible polynomial is a polynomial (of degree greater than zero) that cannot be factored into polynomials of lower degrees. The irreducibility property depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors should belong.
In this way we have that:
[tex]x^3 - 27y^6[/tex]
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Final answer:
None of the given options is a prime polynomial. All the presented polynomials can be factored, so none of them are prime. A prime polynomial is a non-zero polynomial that cannot be factored into other non-constant polynomials.
Explanation:
The subject you've asked about relates to prime polynomials in Mathematics. A prime polynomial over a field is a non-zero polynomial that cannot be factored into the product of other non-constant polynomials from that same field. The provided options are:
- x^3 - 27y^6
- x^4 + 20x^2 - 100
- 3x^2 + 18y
- 10x^4 - 5x^3 + 70x^2 + 3x
In this case, none of the given polynomials is a prime polynomial, as they can all be factored. For instance, x^3 - 27y^6 is a difference of cubes which can be factored, x^4 + 20x^2 - 100 is quadratic in form which can also be factored, 3x^2 + 18y can factored by taking 3 as common, and 10x^4 - 5x^3 + 70x^2 +3x can also be factored.
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