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Answer :
To determine which expression is a prime polynomial, we need to check each polynomial to see if it can be factored further apart from factoring out common factors.
A prime polynomial is one that cannot be factored into two or more nontrivial polynomials over the integers.
Let's go through each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:
- This polynomial does not have any common factors that can be factored out.
- The expression cannot be factored further into simpler polynomials with integer coefficients, making it a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]:
- We notice that both terms have a common factor of 3.
- Factoring out the common factor: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored into simpler terms with integer coefficients, this is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]:
- This is a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
- Using the difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
- This can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]:
- This expression is quadratic in form (think of [tex]\(x^2\)[/tex] as the variable).
- It can be factored into two binomials: [tex]\((x^2 + 10)(x^2 - 10)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
After reviewing each option, we can conclude that:
- Option A ([tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]) is a prime polynomial, as it cannot be factored further apart from factoring out common factors and remains non-factorable over the integers.
A prime polynomial is one that cannot be factored into two or more nontrivial polynomials over the integers.
Let's go through each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:
- This polynomial does not have any common factors that can be factored out.
- The expression cannot be factored further into simpler polynomials with integer coefficients, making it a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]:
- We notice that both terms have a common factor of 3.
- Factoring out the common factor: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored into simpler terms with integer coefficients, this is not a prime polynomial.
C. [tex]\(x^3 - 27y^6\)[/tex]:
- This is a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
- Using the difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
- This can be factored as [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]:
- This expression is quadratic in form (think of [tex]\(x^2\)[/tex] as the variable).
- It can be factored into two binomials: [tex]\((x^2 + 10)(x^2 - 10)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
After reviewing each option, we can conclude that:
- Option A ([tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]) is a prime polynomial, as it cannot be factored further apart from factoring out common factors and remains non-factorable over the integers.
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