Answer :
To determine which expression is a prime polynomial, we need to understand what a prime polynomial is. A prime polynomial is one that cannot be factored into simpler polynomials over the set of integers (or within its coefficient field).
Let's analyze each option:
A. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression can be recognized as a difference of cubes: [tex]\( a^3 - b^3 \)[/tex], where [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex].
- It can be factored using the formula: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex].
- So, [tex]\( x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \)[/tex].
- This means it is not a prime polynomial because it can be factored.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the greatest common factor (GCF) which is [tex]\( x \)[/tex]: [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- The expression inside the parenthesis may not factor further over integers, but since we factored out [tex]\( x \)[/tex], this means the original expression is not prime.
C. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- Let's try factoring this expression using substitution. Let [tex]\( u = x^2 \)[/tex], so the expression becomes [tex]\( u^2 + 20u - 100 \)[/tex].
- We can factor this quadratic as [tex]\( (u + 30)(u - 10) \)[/tex].
- Substituting back for [tex]\( x^2 \)[/tex], we have [tex]\( (x^2 + 30)(x^2 - 10) \)[/tex].
- This means it is not a prime polynomial because it can be factored.
D. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the GCF, which is 3: [tex]\( 3(x^2 + 6y) \)[/tex].
- The presence of a factor (3) indicates that the expression is not prime.
Conclusion:
All the given expressions can be factored, meaning none are prime polynomials. However, there might be an error in the options as typically the answer would be one that could not be factored, but based on the polynomials presented, none fit the criteria for being prime over integers. If this happens on an exam, it would be best to consult with a teacher or verify if there was an omission or mistake in the problem itself.
Let's analyze each option:
A. [tex]\( x^3 - 27y^6 \)[/tex]
- This expression can be recognized as a difference of cubes: [tex]\( a^3 - b^3 \)[/tex], where [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex].
- It can be factored using the formula: [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex].
- So, [tex]\( x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \)[/tex].
- This means it is not a prime polynomial because it can be factored.
B. [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
- We can factor out the greatest common factor (GCF) which is [tex]\( x \)[/tex]: [tex]\( x(10x^3 - 5x^2 + 70x + 3) \)[/tex].
- The expression inside the parenthesis may not factor further over integers, but since we factored out [tex]\( x \)[/tex], this means the original expression is not prime.
C. [tex]\( x^4 + 20x^2 - 100 \)[/tex]
- Let's try factoring this expression using substitution. Let [tex]\( u = x^2 \)[/tex], so the expression becomes [tex]\( u^2 + 20u - 100 \)[/tex].
- We can factor this quadratic as [tex]\( (u + 30)(u - 10) \)[/tex].
- Substituting back for [tex]\( x^2 \)[/tex], we have [tex]\( (x^2 + 30)(x^2 - 10) \)[/tex].
- This means it is not a prime polynomial because it can be factored.
D. [tex]\( 3x^2 + 18y \)[/tex]
- We can factor out the GCF, which is 3: [tex]\( 3(x^2 + 6y) \)[/tex].
- The presence of a factor (3) indicates that the expression is not prime.
Conclusion:
All the given expressions can be factored, meaning none are prime polynomials. However, there might be an error in the options as typically the answer would be one that could not be factored, but based on the polynomials presented, none fit the criteria for being prime over integers. If this happens on an exam, it would be best to consult with a teacher or verify if there was an omission or mistake in the problem itself.
