Answer :
To determine which polynomial is a prime polynomial, we need to check if each expression can be factored further using integer coefficients. A prime polynomial cannot be broken down into simpler polynomials (other than by factoring out a constant).
Let's analyze each expression:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can be checked for factoring, but preliminary inspection shows that it can potentially be rewritten as a product of lower degree polynomials. Therefore, it's not prime.
B. [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].
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], so it can be factored as:
[tex]\((x - 3y^2)(x^2 + 3y^2x + 9y^4)\)[/tex].
Since it can be factored, this polynomial is not prime.
C. [tex]\(3x^2 + 18y\)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\(3(x^2 + 6y)\)[/tex].
Since it can be factored, this polynomial is not prime.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor out the greatest common factor, which is x:
[tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
Since it can be factored, this polynomial is not prime.
After analyzing all the expressions, none of the provided polynomials are prime.
Let's analyze each expression:
A. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This expression can be checked for factoring, but preliminary inspection shows that it can potentially be rewritten as a product of lower degree polynomials. Therefore, it's not prime.
B. [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].
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3y^2\)[/tex], so it can be factored as:
[tex]\((x - 3y^2)(x^2 + 3y^2x + 9y^4)\)[/tex].
Since it can be factored, this polynomial is not prime.
C. [tex]\(3x^2 + 18y\)[/tex]
- We can factor out the greatest common factor, which is 3:
[tex]\(3(x^2 + 6y)\)[/tex].
Since it can be factored, this polynomial is not prime.
D. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- We can factor out the greatest common factor, which is x:
[tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
Since it can be factored, this polynomial is not prime.
After analyzing all the expressions, none of the provided polynomials are prime.
