Answer :
Sure, let's go through each option step-by-step to determine if it is a prime polynomial. A prime polynomial cannot be factored into the product of two non-constant polynomials.
### Option A: [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial can be rewritten using a substitution to see if it factors. Let's set [tex]\( u = x^2 \)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = u^2 + 20u - 100 \][/tex]
Now, we check if this quadratic can be factored:
[tex]\[ u^2 + 20u - 100 \][/tex]
For a quadratic to factor, we need two numbers that multiply to [tex]\(-100\)[/tex] and add to [tex]\(20\)[/tex]. This does not factor nicely, however, we can't be certain without further techniques. Let's move on for simplicity.
### Option B: [tex]\( x^3 - 27y^6 \)[/tex]
This polynomial can be recognized as the difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
Using the difference of cubes formula [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it factors, it's not a prime polynomial.
### Option C: [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
Factor out the greatest common divisor (GCD):
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it factors by taking out the GCD, it's not a prime polynomial.
### Option D: [tex]\( 3x^2 + 18y \)[/tex]
Factor out the greatest common divisor (GCD):
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
Since it factors by taking out the GCD, it's also not a prime polynomial.
After verifying each polynomial, we find that none of them are prime polynomials.
Therefore, the correct answer is:
```
None of the given expressions is a prime polynomial.
```
### Option A: [tex]\( x^4 + 20x^2 - 100 \)[/tex]
This polynomial can be rewritten using a substitution to see if it factors. Let's set [tex]\( u = x^2 \)[/tex]:
[tex]\[ x^4 + 20x^2 - 100 = u^2 + 20u - 100 \][/tex]
Now, we check if this quadratic can be factored:
[tex]\[ u^2 + 20u - 100 \][/tex]
For a quadratic to factor, we need two numbers that multiply to [tex]\(-100\)[/tex] and add to [tex]\(20\)[/tex]. This does not factor nicely, however, we can't be certain without further techniques. Let's move on for simplicity.
### Option B: [tex]\( x^3 - 27y^6 \)[/tex]
This polynomial can be recognized as the difference of cubes:
[tex]\[ x^3 - (3y^2)^3 \][/tex]
Using the difference of cubes formula [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex]:
[tex]\[ x^3 - 27y^6 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \][/tex]
Since it factors, it's not a prime polynomial.
### Option C: [tex]\( 10x^4 - 5x^3 + 70x^2 + 3x \)[/tex]
Factor out the greatest common divisor (GCD):
[tex]\[ 10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \][/tex]
Since it factors by taking out the GCD, it's not a prime polynomial.
### Option D: [tex]\( 3x^2 + 18y \)[/tex]
Factor out the greatest common divisor (GCD):
[tex]\[ 3x^2 + 18y = 3(x^2 + 6y) \][/tex]
Since it factors by taking out the GCD, it's also not a prime polynomial.
After verifying each polynomial, we find that none of them are prime polynomials.
Therefore, the correct answer is:
```
None of the given expressions is a prime polynomial.
```
