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Answer :
Sure, let's determine which expression among the given options is a prime polynomial.
First, let's recall what a prime polynomial is. A prime polynomial has no factors other than 1 and itself, meaning it cannot be factored into simpler polynomials with coefficients in the set of integers.
Let's consider each option one by one:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be factored further.
We can check if there are any simpler forms, such as factoring by grouping or other factoring techniques. After analyzing, we determine that it can be factored, so it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
We can see a common factor in the terms:
[tex]\[3x^2 + 18y = 3(x^2 + 6y)\][/tex]
Thus, this is not a prime polynomial because it can be factored.
C. [tex]\(x^3 - 27y^6\)[/tex]
This is a difference of cubes:
[tex]\[x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\][/tex]
Since it can be factored further, this is not a prime polynomial.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial has a common factor in the terms:
[tex]\[10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)\][/tex]
Hence, it is not a prime polynomial because it can be factored.
After analyzing all the options, none of these are prime polynomials.
The answer is none of these polynomials are prime.
First, let's recall what a prime polynomial is. A prime polynomial has no factors other than 1 and itself, meaning it cannot be factored into simpler polynomials with coefficients in the set of integers.
Let's consider each option one by one:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial can be factored further.
We can check if there are any simpler forms, such as factoring by grouping or other factoring techniques. After analyzing, we determine that it can be factored, so it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
We can see a common factor in the terms:
[tex]\[3x^2 + 18y = 3(x^2 + 6y)\][/tex]
Thus, this is not a prime polynomial because it can be factored.
C. [tex]\(x^3 - 27y^6\)[/tex]
This is a difference of cubes:
[tex]\[x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4)\][/tex]
Since it can be factored further, this is not a prime polynomial.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
This polynomial has a common factor in the terms:
[tex]\[10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3)\][/tex]
Hence, it is not a prime polynomial because it can be factored.
After analyzing all the options, none of these are prime polynomials.
The answer is none of these polynomials are prime.
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