Answer :
To determine which of the given expressions is a prime polynomial, we need to analyze each option and see if they can be factored. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
Let's go through each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression can be thought of as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let [tex]\( y = x^2 \)[/tex], then it becomes [tex]\( y^2 + 20y - 100 \)[/tex].
- This is a standard quadratic expression that can be factored, indicating that the original polynomial is not prime.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- First, we can factor out the greatest common factor, which is [tex]\( x \)[/tex], resulting in [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- Since part of the expression can be factored out, the polynomial is not prime.
C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is [tex]\( 3 \)[/tex], resulting in [tex]\( 3(x^2 + 6y) \)[/tex].
- Since the expression can be factored, it's clear that this polynomial is not prime.
D. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes, because [tex]\( x^3 \)[/tex] and [tex]\( (3y^2)^3 \)[/tex] are perfect cubes.
- It can be factored using the formula for the difference of cubes:
[tex]\[
x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
After analyzing each expression, none of them qualify as an irreducible or prime polynomial because all can be factored further into polynomials of lower degree. Therefore, none of the options provided is a prime polynomial.
Let's go through each option:
A. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression can be thought of as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let [tex]\( y = x^2 \)[/tex], then it becomes [tex]\( y^2 + 20y - 100 \)[/tex].
- This is a standard quadratic expression that can be factored, indicating that the original polynomial is not prime.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- First, we can factor out the greatest common factor, which is [tex]\( x \)[/tex], resulting in [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- Since part of the expression can be factored out, the polynomial is not prime.
C. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the greatest common factor, which is [tex]\( 3 \)[/tex], resulting in [tex]\( 3(x^2 + 6y) \)[/tex].
- Since the expression can be factored, it's clear that this polynomial is not prime.
D. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression is a difference of cubes, because [tex]\( x^3 \)[/tex] and [tex]\( (3y^2)^3 \)[/tex] are perfect cubes.
- It can be factored using the formula for the difference of cubes:
[tex]\[
x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
After analyzing each expression, none of them qualify as an irreducible or prime polynomial because all can be factored further into polynomials of lower degree. Therefore, none of the options provided is a prime polynomial.
