Answer :
To determine which expression is a prime polynomial, we need to see if any of the given polynomials can be factored into simpler polynomials with integer coefficients. A prime polynomial is one that cannot be factored further.
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
This shows that the polynomial is not prime.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression looks like a difference of cubes. It can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- To factor this, notice that it can be seen as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100 \][/tex]
By finding roots or attempting factorization, it can be decomposed further. Hence, it is not prime.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression does not have an obvious factoring pattern or common factor other than 1. After examining it, there's no simplification possible with integer coefficients, suggesting that it cannot be factored further. Thus, this polynomial is prime.
Based on this analysis, the expression that is a prime polynomial is option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex].
Let's analyze each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3:
[tex]\[ 3(x^2 + 6y) \][/tex]
This shows that the polynomial is not prime.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This expression looks like a difference of cubes. It can be factored as:
[tex]\[ (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it can be factored, it is not a prime polynomial.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- To factor this, notice that it can be seen as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = (x^2)^2 + 20(x^2) - 100 \][/tex]
By finding roots or attempting factorization, it can be decomposed further. Hence, it is not prime.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This expression does not have an obvious factoring pattern or common factor other than 1. After examining it, there's no simplification possible with integer coefficients, suggesting that it cannot be factored further. Thus, this polynomial is prime.
Based on this analysis, the expression that is a prime polynomial is option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex].
