Answer :
Let's analyze each of the given expressions to determine if it is a prime polynomial. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
Option A: [tex]\( x^3 - 27y^6 \)[/tex]
This expression can be recognized as a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
According to the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
we have [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex]. Therefore:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
Option B: [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
We can factor out an [tex]\( x \)[/tex] from this expression:
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, it is not a prime polynomial.
Option C: [tex]\( 3x^2 + 18y \)[/tex]
We can factor out a 3 from this expression:
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
Option D: [tex]\( x^4 + 20x^2 - 100 \)[/tex]
To determine if this is a prime polynomial, let's attempt to factor it:
Let [tex]\( z = x^2 \)[/tex], then the expression becomes:
[tex]\[ z^2 + 20z - 100 \][/tex]
We look for two numbers that multiply to [tex]\(-100\)[/tex] and add up to [tex]\(20\)[/tex]. After trying possible factors, we fail to find a pair that works, which suggests it might be difficult or impossible to factor further using integer coefficients.
Since option D cannot be factored easily with integer coefficients, it is prime.
Therefore, the answer is D [tex]\( x^4 + 20x^2 - 100 \)[/tex], which is a prime polynomial.
Option A: [tex]\( x^3 - 27y^6 \)[/tex]
This expression can be recognized as a difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
According to the difference of cubes formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
we have [tex]\( a = x \)[/tex] and [tex]\( b = 3y^2 \)[/tex]. Therefore:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
Option B: [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
We can factor out an [tex]\( x \)[/tex] from this expression:
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored, it is not a prime polynomial.
Option C: [tex]\( 3x^2 + 18y \)[/tex]
We can factor out a 3 from this expression:
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
Since it can be factored, it is not a prime polynomial.
Option D: [tex]\( x^4 + 20x^2 - 100 \)[/tex]
To determine if this is a prime polynomial, let's attempt to factor it:
Let [tex]\( z = x^2 \)[/tex], then the expression becomes:
[tex]\[ z^2 + 20z - 100 \][/tex]
We look for two numbers that multiply to [tex]\(-100\)[/tex] and add up to [tex]\(20\)[/tex]. After trying possible factors, we fail to find a pair that works, which suggests it might be difficult or impossible to factor further using integer coefficients.
Since option D cannot be factored easily with integer coefficients, it is prime.
Therefore, the answer is D [tex]\( x^4 + 20x^2 - 100 \)[/tex], which is a prime polynomial.
