Answer :
To determine which expression is a prime polynomial, let's first understand what a prime polynomial is. A prime polynomial is a polynomial that cannot be factored into polynomials of lower degree with integer coefficients.
Now let's analyze each expression:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be tested for factorability. Here, you can look for patterns or try substitution methods, but it's a complex process and typically doesn't result in a prime polynomial of this form.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes. The formula for factoring a difference of cubes [tex]\(a^3 - b^3\)[/tex] is [tex]\((a-b)(a^2+ab+b^2)\)[/tex]. Here, [tex]\(x^3\)[/tex] is [tex]\(a^3\)[/tex] and [tex]\((3y^2)^3\)[/tex] (or [tex]\(27y^6\)[/tex]) is [tex]\(b^3\)[/tex]. It can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
C. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor (GCF), which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
This shows it is not a prime polynomial, as it can be reduced further by factoring out the 3.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial could be factored using grouping or other factoring methods, which means it is not prime. Factoring by grouping, you can attempt to take out common terms or analyze simpler commonalities to reduce it further.
After analyzing each expression, none of them can be considered a prime polynomial because all can be factored further. Thus, none of the given expressions is a prime polynomial.
Now let's analyze each expression:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be tested for factorability. Here, you can look for patterns or try substitution methods, but it's a complex process and typically doesn't result in a prime polynomial of this form.
B. [tex]\(x^3 - 27y^6\)[/tex]
This expression is a difference of cubes. The formula for factoring a difference of cubes [tex]\(a^3 - b^3\)[/tex] is [tex]\((a-b)(a^2+ab+b^2)\)[/tex]. Here, [tex]\(x^3\)[/tex] is [tex]\(a^3\)[/tex] and [tex]\((3y^2)^3\)[/tex] (or [tex]\(27y^6\)[/tex]) is [tex]\(b^3\)[/tex]. It can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
C. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor (GCF), which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
This shows it is not a prime polynomial, as it can be reduced further by factoring out the 3.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial could be factored using grouping or other factoring methods, which means it is not prime. Factoring by grouping, you can attempt to take out common terms or analyze simpler commonalities to reduce it further.
After analyzing each expression, none of them can be considered a prime polynomial because all can be factored further. Thus, none of the given expressions is a prime polynomial.
