Answer :
To determine which expression is a prime polynomial, we need to examine each option and see if it can be factored further or not. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
Let's look at each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression is already in its simplest form and cannot be factored further using integer coefficients. Thus, it is a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression can be factored further since it has a common factor of [tex]\(x\)[/tex], and possibly more additionally.
C. [tex]\(x^3 - 27y^6\)[/tex]
- This expression can be recognized as a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex]. It can be factored using the difference of cubes formula.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression has a common factor of [tex]\(3\)[/tex], which can be factored out, simplifying it.
After analyzing each option, we conclude that option A, [tex]\(x^4 + 20x^2 - 100\)[/tex], is a prime polynomial as it cannot be factored further using integer coefficients.
Let's look at each option:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression is already in its simplest form and cannot be factored further using integer coefficients. Thus, it is a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression can be factored further since it has a common factor of [tex]\(x\)[/tex], and possibly more additionally.
C. [tex]\(x^3 - 27y^6\)[/tex]
- This expression can be recognized as a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex]. It can be factored using the difference of cubes formula.
D. [tex]\(3x^2 + 18y\)[/tex]
- This expression has a common factor of [tex]\(3\)[/tex], which can be factored out, simplifying it.
After analyzing each option, we conclude that option A, [tex]\(x^4 + 20x^2 - 100\)[/tex], is a prime polynomial as it cannot be factored further using integer coefficients.
