Answer :
To determine which expression is a prime polynomial, we need to understand what a prime polynomial is. A prime polynomial is a polynomial that cannot be factored into polynomials of lower degree with integer coefficients other than itself and one.
Let's go through each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by factoring out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored further, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression 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]. Here, [tex]\(a = x\)[/tex] and [tex]\(b = (3y^2)\)[/tex]:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial can be factored by taking out the common factor of [tex]\(x\)[/tex]:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored further, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial can be tricky to factor. However, upon inspection, it does not factor into simpler polynomials with integer coefficients, which suggests it might be a prime polynomial. It remains unfactored using common rational factorization techniques.
Therefore, the expression that is a prime polynomial is likely option D, [tex]\(x^4 + 20x^2 - 100\)[/tex].
Let's go through each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial can be factored by factoring out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
Since it can be factored further, it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression 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]. Here, [tex]\(a = x\)[/tex] and [tex]\(b = (3y^2)\)[/tex]:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial can be factored by taking out the common factor of [tex]\(x\)[/tex]:
[tex]\[ x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it can be factored further, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial can be tricky to factor. However, upon inspection, it does not factor into simpler polynomials with integer coefficients, which suggests it might be a prime polynomial. It remains unfactored using common rational factorization techniques.
Therefore, the expression that is a prime polynomial is likely option D, [tex]\(x^4 + 20x^2 - 100\)[/tex].
