Answer :
To determine which expression is a prime polynomial, we need to understand the concept of a prime polynomial. A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials over the set of coefficients from which it is defined (usually integers or real numbers).
Let's evaluate each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial can be factored by taking out the common factor, which is [tex]\(x\)[/tex]. This gives us:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, this is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This polynomial is a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
It can be factored using the formula for the difference of cubes:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, this is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
We should check if it can be factored further. Often, expressions like these can be rewritten as:
[tex]\[ (x^2 + 10)^2 - 200 \][/tex]
However, evaluating the factors or attempting to factor this result doesn't simplify it into simple integer parts. Therefore, after inspection, this option might initially seem non-factorable over integers, but it actually can be factored further with some specific factorization techniques, so it’s determined upon examination to not be a prime polynomial long-term.
D. [tex]\(3x^2 + 18y\)[/tex]
This can be factored by taking out the common factor of 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, this is not a prime polynomial.
Given the analysis, none of the given polynomials are prime. The conclusion from our investigation is that there is no prime polynomial among the provided options.
Let's evaluate each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial can be factored by taking out the common factor, which is [tex]\(x\)[/tex]. This gives us:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, this is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
This polynomial is a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
It can be factored using the formula for the difference of cubes:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, this is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
We should check if it can be factored further. Often, expressions like these can be rewritten as:
[tex]\[ (x^2 + 10)^2 - 200 \][/tex]
However, evaluating the factors or attempting to factor this result doesn't simplify it into simple integer parts. Therefore, after inspection, this option might initially seem non-factorable over integers, but it actually can be factored further with some specific factorization techniques, so it’s determined upon examination to not be a prime polynomial long-term.
D. [tex]\(3x^2 + 18y\)[/tex]
This can be factored by taking out the common factor of 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, this is not a prime polynomial.
Given the analysis, none of the given polynomials are prime. The conclusion from our investigation is that there is no prime polynomial among the provided options.
