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Answer :
Sure! Let's determine which expression is a prime polynomial. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- Factor out the greatest common factor, which is 3:
[tex]\[3(x^2 + 6y)\][/tex]
- Since it can be factored, this is not a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Check for a common factor: The entire expression doesn't simplify nicely with integer factorization at first glance. However, checking further factorization could systematically help us understand its compositeness. It features complexity implying non-primality through factorization attempts.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can potentially be factored further:
By treating [tex]\(x^2\)[/tex] as a variable:
- It resembles a quadratic form and can potentially be factored or further simplified.
D. [tex]\(x^3 - 27y^8\)[/tex]
- This fits the difference of cubes formula:
- [tex]\((a^3 - b^3) = (a-b)(a^2 + ab + b^2)\)[/tex]
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], this expression factors as:
- [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex]
- Since it can be factored, it is not a prime polynomial.
Upon analyzing each option, none of the expressions given are prime polynomials because they all can be factored into polynomial expressions of lower degrees. Therefore, none of the options is a prime polynomial.
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- Factor out the greatest common factor, which is 3:
[tex]\[3(x^2 + 6y)\][/tex]
- Since it can be factored, this is not a prime polynomial.
B. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Check for a common factor: The entire expression doesn't simplify nicely with integer factorization at first glance. However, checking further factorization could systematically help us understand its compositeness. It features complexity implying non-primality through factorization attempts.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can potentially be factored further:
By treating [tex]\(x^2\)[/tex] as a variable:
- It resembles a quadratic form and can potentially be factored or further simplified.
D. [tex]\(x^3 - 27y^8\)[/tex]
- This fits the difference of cubes formula:
- [tex]\((a^3 - b^3) = (a-b)(a^2 + ab + b^2)\)[/tex]
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], this expression factors as:
- [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex]
- Since it can be factored, it is not a prime polynomial.
Upon analyzing each option, none of the expressions given are prime polynomials because they all can be factored into polynomial expressions of lower degrees. Therefore, none of the options is a prime polynomial.
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