Answer :
Sure! Let's analyze each of the given polynomials to determine which one is a prime polynomial. A prime polynomial is a polynomial that can't be factored into the product of two non-constant polynomials with coefficients in the same field.
Here are the polynomials:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial has four terms.
- It does not seem to simplify easily.
- Let's consider the factors; it does not look like this polynomial can be factored into simpler polynomials.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This is a difference of cubes, which can be factored using the identity [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 factors to:
[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.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial resembles a quadratic form if we let [tex]\(z = x^2\)[/tex]: [tex]\(z^2 + 20z - 100\)[/tex].
- We can try to factor it, but it doesn't simplify into the product of two simpler polynomials over the integers.
- This suggests that it may be irreducible.
D. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial has a common factor of 3.
- Factor out the 3:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
From this analysis, the correct answer is:
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial cannot be factored further and is therefore a prime polynomial.
Here are the polynomials:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- This polynomial has four terms.
- It does not seem to simplify easily.
- Let's consider the factors; it does not look like this polynomial can be factored into simpler polynomials.
B. [tex]\(x^3 - 27y^6\)[/tex]
- This is a difference of cubes, which can be factored using the identity [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 factors to:
[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.
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- This polynomial resembles a quadratic form if we let [tex]\(z = x^2\)[/tex]: [tex]\(z^2 + 20z - 100\)[/tex].
- We can try to factor it, but it doesn't simplify into the product of two simpler polynomials over the integers.
- This suggests that it may be irreducible.
D. [tex]\(3x^2 + 18y\)[/tex]
- This polynomial has a common factor of 3.
- Factor out the 3:
[tex]\[
3x^2 + 18y = 3(x^2 + 6y)
\][/tex]
- Since it can be factored, it is not a prime polynomial.
From this analysis, the correct answer is:
C. [tex]\(x^4 + 20x^2 - 100\)[/tex]
This polynomial cannot be factored further and is therefore a prime polynomial.
