Answer :
Sure! To determine which expression is a prime polynomial, we need to understand what a prime polynomial is. A prime polynomial is a polynomial that cannot be factored into the product of two or more non-constant polynomials with coefficients in the same set (typically integers).
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored. Common factors can be factored out:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
It's not a prime polynomial because it can be factored further.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Look for common factors or other factorization. This expression might seem complex, but it doesn't simplify nicely into a product of polynomials with integer coefficients.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can be factored. It is a type of quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[
x^4 + 20x^2 - 100 = (x^2 + 10)^2 - 200 = (x^2 + 10 - \sqrt{200})(x^2 + 10 + \sqrt{200})
\][/tex]
While these factors look complex, the method shows potential factors.
D. [tex]\(x^3 - 27y^6\)[/tex]
- This expression can be factored using the difference of cubes identity:
[tex]\[
x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
It is not a prime polynomial because it can be further factored.
The solution identifies that option A, [tex]\(3x^2 + 18y\)[/tex], is not a prime polynomial due to factoring, although at first glance, it might seem like it is not easily simplified into smaller degrees.
For clarity, note that this method and exploration found that none of the given options are prime polynomials due to being able to factorize them further or simplify in a certain form.
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored. Common factors can be factored out:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
It's not a prime polynomial because it can be factored further.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Look for common factors or other factorization. This expression might seem complex, but it doesn't simplify nicely into a product of polynomials with integer coefficients.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can be factored. It is a type of quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[
x^4 + 20x^2 - 100 = (x^2 + 10)^2 - 200 = (x^2 + 10 - \sqrt{200})(x^2 + 10 + \sqrt{200})
\][/tex]
While these factors look complex, the method shows potential factors.
D. [tex]\(x^3 - 27y^6\)[/tex]
- This expression can be factored using the difference of cubes identity:
[tex]\[
x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
It is not a prime polynomial because it can be further factored.
The solution identifies that option A, [tex]\(3x^2 + 18y\)[/tex], is not a prime polynomial due to factoring, although at first glance, it might seem like it is not easily simplified into smaller degrees.
For clarity, note that this method and exploration found that none of the given options are prime polynomials due to being able to factorize them further or simplify in a certain form.
