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Answer :
To determine which polynomial expression is a prime polynomial, we need to check if each expression can be factored further. A prime polynomial cannot be factored into a product of polynomials with integer coefficients.
Let's review each option:
A. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression represents a difference of cubes: [tex]\( x^3 - (3y^2)^3 \)[/tex].
- It can be factored as: [tex]\((x - 3y^2)(x^2 + 3xy^2 + (3y^2)^2)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
B. [tex]\( 3x^2 + 18y \)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3.
- Factoring gives us: [tex]\( 3(x^2 + 6y) \)[/tex].
- Since it can be factored, it is not a prime polynomial.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out an [tex]\( x \)[/tex] from all terms, resulting in: [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- The expression can be partially factored, so it is not a prime polynomial.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression can be factored as follows:
- Recognize it as a quadratic in form: [tex]\( (x^2)^2 + 20(x^2) - 100 \)[/tex].
- It factors to: [tex]\((x^2 - 10)(x^2 + 10)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
From the analysis above, none of the given expressions are prime polynomials, as they can all be factored.
Let's review each option:
A. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression represents a difference of cubes: [tex]\( x^3 - (3y^2)^3 \)[/tex].
- It can be factored as: [tex]\((x - 3y^2)(x^2 + 3xy^2 + (3y^2)^2)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
B. [tex]\( 3x^2 + 18y \)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3.
- Factoring gives us: [tex]\( 3(x^2 + 6y) \)[/tex].
- Since it can be factored, it is not a prime polynomial.
C. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out an [tex]\( x \)[/tex] from all terms, resulting in: [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- The expression can be partially factored, so it is not a prime polynomial.
D. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- This expression can be factored as follows:
- Recognize it as a quadratic in form: [tex]\( (x^2)^2 + 20(x^2) - 100 \)[/tex].
- It factors to: [tex]\((x^2 - 10)(x^2 + 10)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
From the analysis above, none of the given expressions are prime polynomials, as they can all be factored.
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