Answer :
To determine which expression is a prime polynomial, let's first understand what a prime polynomial is. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
Let’s examine each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial has a common factor of [tex]\(x\)[/tex], so it can be factored at least as [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. Since it can be factored further, it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
This polynomial can be factored by taking out the greatest common factor, which is 3, resulting in [tex]\(3(x^2 + 6y)\)[/tex]. Since it can be factored further, it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be recognized as a difference of cubes, which can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex]. Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression can be factored over the integers. It can be rewritten as [tex]\((x^2 + 10)^2 - 10^2\)[/tex], which is a difference of squares: [tex]\((x^2 + 10 - 10)(x^2 + 10 + 10)\)[/tex]. Thus it factors to [tex]\((x^2 - 0)(x^2 + 20)\)[/tex] and further exploration shows more options over specific number sets. Therefore, it’s not a prime polynomial.
None of the expressions listed is truly prime, but generally on selections like these, further simplifying with integer coefficients shows no prime reduction as noted.
So, based on factoring exploration, expressions were checked for primality by assessing factors and decompositions.
Let’s examine each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial has a common factor of [tex]\(x\)[/tex], so it can be factored at least as [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex]. Since it can be factored further, it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
This polynomial can be factored by taking out the greatest common factor, which is 3, resulting in [tex]\(3(x^2 + 6y)\)[/tex]. Since it can be factored further, it is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be recognized as a difference of cubes, which can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex]. Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This expression can be factored over the integers. It can be rewritten as [tex]\((x^2 + 10)^2 - 10^2\)[/tex], which is a difference of squares: [tex]\((x^2 + 10 - 10)(x^2 + 10 + 10)\)[/tex]. Thus it factors to [tex]\((x^2 - 0)(x^2 + 20)\)[/tex] and further exploration shows more options over specific number sets. Therefore, it’s not a prime polynomial.
None of the expressions listed is truly prime, but generally on selections like these, further simplifying with integer coefficients shows no prime reduction as noted.
So, based on factoring exploration, expressions were checked for primality by assessing factors and decompositions.
