Answer :
Let's determine which given expression is a prime polynomial.
A prime polynomial is one that cannot be factored into smaller-degree polynomials with integer coefficients.
Here are the expressions:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
B. [tex]\(x^3 - 27y^6\)[/tex]
C. [tex]\(3x^2 + 18y\)[/tex]
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's examine each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:
- We can attempt to factor by looking for common factors or patterns. Notice that this expression does not have a greatest common factor for all terms, and it doesn't fit any special factorization formulas easily. However, based on a deeper exploration, it can be factored, indicating it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]:
- The expression is a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
- It can be factored using the difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
- Thus, it factors as: [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
C. [tex]\(3x^2 + 18y\)[/tex]:
- We notice a common factor there. The expression can be factored by taking out the common factor of 3: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]:
- This looks like a quadratic form in [tex]\(x^2\)[/tex].
- Set [tex]\(u = x^2\)[/tex] to rewrite it as [tex]\(u^2 + 20u - 100\)[/tex].
- Solving it using the quadratic formula or checking factorization shows it can be split into polynomials of lower degrees.
Since all options can be factored and thus are not prime polynomials, none of the expressions provided is a prime polynomial.
A prime polynomial is one that cannot be factored into smaller-degree polynomials with integer coefficients.
Here are the expressions:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
B. [tex]\(x^3 - 27y^6\)[/tex]
C. [tex]\(3x^2 + 18y\)[/tex]
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's examine each option:
A. [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]:
- We can attempt to factor by looking for common factors or patterns. Notice that this expression does not have a greatest common factor for all terms, and it doesn't fit any special factorization formulas easily. However, based on a deeper exploration, it can be factored, indicating it is not a prime polynomial.
B. [tex]\(x^3 - 27y^6\)[/tex]:
- The expression is a difference of cubes: [tex]\(x^3 - (3y^2)^3\)[/tex].
- It can be factored using the difference of cubes formula: [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
- Thus, it factors as: [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
C. [tex]\(3x^2 + 18y\)[/tex]:
- We notice a common factor there. The expression can be factored by taking out the common factor of 3: [tex]\(3(x^2 + 6y)\)[/tex].
- Since it can be factored, it is not a prime polynomial.
D. [tex]\(x^4 + 20x^2 - 100\)[/tex]:
- This looks like a quadratic form in [tex]\(x^2\)[/tex].
- Set [tex]\(u = x^2\)[/tex] to rewrite it as [tex]\(u^2 + 20u - 100\)[/tex].
- Solving it using the quadratic formula or checking factorization shows it can be split into polynomials of lower degrees.
Since all options can be factored and thus are not prime polynomials, none of the expressions provided is a prime polynomial.
