Answer :
To determine which expression is a prime polynomial, let's examine each option to see if they can be factored into simpler polynomials. A prime polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients.
Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's see if this polynomial can be factored:
- Break down the expression:
- [tex]\(x^4 + 20x^2 - 100\)[/tex] can be written in terms of quadratic form as [tex]\((x^2)^2 + 20(x^2) - 100\)[/tex].
- This can potentially factor as [tex]\((x^2 - 10)(x^2 + 10)\)[/tex].
Since it can be factored, it's not a prime polynomial.
Option B: [tex]\(3x^2 + 18y\)[/tex]
Let's check if it's factorable:
- Notice that both terms share a common factor of 3:
- Factoring out the 3 gives us: [tex]\(3(x^2 + 6y)\)[/tex].
Since it can be factored, this is not a prime polynomial.
Option C: [tex]\(x^3 - 27y^6\)[/tex]
Let's examine its factorability:
- Recognize this as a difference of cubes:
- The expression can be rewritten as [tex]\((x)^3 - (3y^2)^3\)[/tex].
- The difference of cubes formula is [tex]\(a^3 - b^3 = (a-b)(a^2+ab+b^2)\)[/tex].
- Applying the formula gives us: [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
Since this can be factored, it’s not a prime polynomial.
Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Let's review its factorability:
- Note that every term has an [tex]\(x\)[/tex] and can be factored by extracting the greatest common factor:
- Factoring out [tex]\(x\)[/tex] yields: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
Since it can be factored, it's not a prime polynomial.
Conclusion:
Each option can be factored, meaning none of these expressions is a prime polynomial. Thus, none of the options provided is a prime polynomial.
Option A: [tex]\(x^4 + 20x^2 - 100\)[/tex]
Let's see if this polynomial can be factored:
- Break down the expression:
- [tex]\(x^4 + 20x^2 - 100\)[/tex] can be written in terms of quadratic form as [tex]\((x^2)^2 + 20(x^2) - 100\)[/tex].
- This can potentially factor as [tex]\((x^2 - 10)(x^2 + 10)\)[/tex].
Since it can be factored, it's not a prime polynomial.
Option B: [tex]\(3x^2 + 18y\)[/tex]
Let's check if it's factorable:
- Notice that both terms share a common factor of 3:
- Factoring out the 3 gives us: [tex]\(3(x^2 + 6y)\)[/tex].
Since it can be factored, this is not a prime polynomial.
Option C: [tex]\(x^3 - 27y^6\)[/tex]
Let's examine its factorability:
- Recognize this as a difference of cubes:
- The expression can be rewritten as [tex]\((x)^3 - (3y^2)^3\)[/tex].
- The difference of cubes formula is [tex]\(a^3 - b^3 = (a-b)(a^2+ab+b^2)\)[/tex].
- Applying the formula gives us: [tex]\((x - 3y^2)(x^2 + 3xy^2 + 9y^4)\)[/tex].
Since this can be factored, it’s not a prime polynomial.
Option D: [tex]\(10x^4 - 5x^3 + 70x^2 + 3x\)[/tex]
Let's review its factorability:
- Note that every term has an [tex]\(x\)[/tex] and can be factored by extracting the greatest common factor:
- Factoring out [tex]\(x\)[/tex] yields: [tex]\(x(10x^3 - 5x^2 + 70x + 3)\)[/tex].
Since it can be factored, it's not a prime polynomial.
Conclusion:
Each option can be factored, meaning none of these expressions is a prime polynomial. Thus, none of the options provided is a prime polynomial.
