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Answer :
To determine which expression is a prime polynomial, we need to see if any of these polynomials can be factored further. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with coefficients in the same field.
Let's analyze each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be written as a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
The formula for factoring a difference of cubes is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
So, we can factor [tex]\(x^3 - 27y^6\)[/tex] as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
This expression can be factored, so it's not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This is a quadratic in form, where [tex]\(x^2\)[/tex] is the variable, say [tex]\(u = x^2\)[/tex]. Substituting, we have:
[tex]\[ u^2 + 20u - 100 \][/tex]
This can be factored using the quadratic formula or by finding factors:
[tex]\[ (u + 10)(u - 10) = (x^2 + 10)(x^2 - 10) \][/tex]
This expression is factorable, so it's not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
We can try factoring out a common factor from the terms:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since this expression can be factored by taking out an [tex]\(x\)[/tex], it's not a prime polynomial.
D. [tex]\(3x^2 + 18y\)[/tex]
This expression can have a common factor factored out:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since we can factor out the constant 3, this polynomial is not prime.
None of the expressions given are prime as they all can be factored. It seems there might be a misunderstanding or error in listing the choices. Let me evaluate again more thoroughly because the factored form implies possible errors in the options for a prime expression since none technically fulfill the criterion of being unable to factor once factoring is performed accurately. Please cross-check for any errors or additional context changes not covered.
Let's analyze each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This expression can be written as a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
The formula for factoring a difference of cubes is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
So, we can factor [tex]\(x^3 - 27y^6\)[/tex] as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
This expression can be factored, so it's not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This is a quadratic in form, where [tex]\(x^2\)[/tex] is the variable, say [tex]\(u = x^2\)[/tex]. Substituting, we have:
[tex]\[ u^2 + 20u - 100 \][/tex]
This can be factored using the quadratic formula or by finding factors:
[tex]\[ (u + 10)(u - 10) = (x^2 + 10)(x^2 - 10) \][/tex]
This expression is factorable, so it's not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
We can try factoring out a common factor from the terms:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since this expression can be factored by taking out an [tex]\(x\)[/tex], it's not a prime polynomial.
D. [tex]\(3x^2 + 18y\)[/tex]
This expression can have a common factor factored out:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since we can factor out the constant 3, this polynomial is not prime.
None of the expressions given are prime as they all can be factored. It seems there might be a misunderstanding or error in listing the choices. Let me evaluate again more thoroughly because the factored form implies possible errors in the options for a prime expression since none technically fulfill the criterion of being unable to factor once factoring is performed accurately. Please cross-check for any errors or additional context changes not covered.
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