Answer :
To determine which of the given expressions is a prime polynomial, we need to check if any of them cannot be factored into simpler polynomials with integer coefficients.
Here's the breakdown for each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This is a difference of cubes, which can be factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this expression, [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex]:
[tex]\[ x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
First, factor out the greatest common factor:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
This further factoring is complex and checking individually can show it's not easily factored into smaller polynomials over the integers. So, this might seem complicated but requires a deeper look to confirm primality.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This is a quadratic in form if [tex]\( x^2\)[/tex] is substituted as a single term:
Let [tex]\( z = x^2\)[/tex]:
[tex]\[ z^2 + 20z - 100 \][/tex]
We can attempt to factor this quadratic:
[tex]\[ (z + 30)(z - 10) \][/tex]
Converting back:
[tex]\[ (x^2 + 30)(x^2 - 10) \][/tex]
Since it can be factored, it is not a prime polynomial.
By analyzing the options, option C is the one less straightforward in factoring and thus might be a candidate for being a prime polynomial when complex factorization methods show it doesn't break into simpler terms easily. However, for typical cases and simplification, the list here mostly points it not terminally factored as assumptions without deeper techniques.
Conclusively, with fundamental checking, option A, B, and D break down while option A is not distinct, larger study can define C, thus among common check when resources rather support C atypical support.
Here's the breakdown for each option:
A. [tex]\(x^3 - 27y^6\)[/tex]
This is a difference of cubes, which can be factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this expression, [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex]:
[tex]\[ x^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
B. [tex]\(3x^2 + 18y\)[/tex]
This expression can be factored by taking out the greatest common factor:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
First, factor out the greatest common factor:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
This further factoring is complex and checking individually can show it's not easily factored into smaller polynomials over the integers. So, this might seem complicated but requires a deeper look to confirm primality.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This is a quadratic in form if [tex]\( x^2\)[/tex] is substituted as a single term:
Let [tex]\( z = x^2\)[/tex]:
[tex]\[ z^2 + 20z - 100 \][/tex]
We can attempt to factor this quadratic:
[tex]\[ (z + 30)(z - 10) \][/tex]
Converting back:
[tex]\[ (x^2 + 30)(x^2 - 10) \][/tex]
Since it can be factored, it is not a prime polynomial.
By analyzing the options, option C is the one less straightforward in factoring and thus might be a candidate for being a prime polynomial when complex factorization methods show it doesn't break into simpler terms easily. However, for typical cases and simplification, the list here mostly points it not terminally factored as assumptions without deeper techniques.
Conclusively, with fundamental checking, option A, B, and D break down while option A is not distinct, larger study can define C, thus among common check when resources rather support C atypical support.
