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Answer :
To determine which expression is a prime polynomial, we need to analyze each choice. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients (or the coefficients of the given polynomials).
Let's go through each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3.
- Factoring gives us: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- We can simplify our approach by letting [tex]\(z = x^2\)[/tex], transforming the polynomial into [tex]\(z^2 + 20z - 100\)[/tex].
- This quadratic can be solved or factored further, but since it has potential factors, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Observe that there is a common factor of [tex]\(x\)[/tex] in all the terms.
- Factoring out the common factor gives us: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since there is a common factor of [tex]\(x\)[/tex], it is not a prime polynomial.
D. [tex]\(x^3 - 27y^6\)[/tex]
- Recognize this as a difference of cubes, where [tex]\(x^3\)[/tex] and [tex]\((3y^2)^3\)[/tex] are perfect cubes.
- The difference of cubes can be factored into: [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored using the difference of cubes formula, it is not a prime polynomial.
After evaluating each option, we find that none of the given polynomials are prime.
Let's go through each option:
A. [tex]\(3x^2 + 18y\)[/tex]
- This expression can be factored by taking out the greatest common factor, which is 3.
- Factoring gives us: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
B. [tex]\(x^4 + 20x^2 - 100\)[/tex]
- We can simplify our approach by letting [tex]\(z = x^2\)[/tex], transforming the polynomial into [tex]\(z^2 + 20z - 100\)[/tex].
- This quadratic can be solved or factored further, but since it has potential factors, it is not a prime polynomial.
C. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
- Observe that there is a common factor of [tex]\(x\)[/tex] in all the terms.
- Factoring out the common factor gives us: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
- Since there is a common factor of [tex]\(x\)[/tex], it is not a prime polynomial.
D. [tex]\(x^3 - 27y^6\)[/tex]
- Recognize this as a difference of cubes, where [tex]\(x^3\)[/tex] and [tex]\((3y^2)^3\)[/tex] are perfect cubes.
- The difference of cubes can be factored into: [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
- Since it can be factored using the difference of cubes formula, it is not a prime polynomial.
After evaluating each option, we find that none of the given polynomials are prime.
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