To determine which polynomials are prime, we need to examine each polynomial and see if it can be factored into the product of polynomials of lower degree with coefficients in the same field (usually the field of real numbers or complex numbers). A polynomial is considered prime (also called irreducible) if it cannot be factored in such a way.
Let's examine each polynomial individually:
1. [tex]\( x^2 + 9 \)[/tex]
- This polynomial is [tex]\( x^2 + 9 \)[/tex]. To check if it's prime, we need to see if it can be factored. Over the real numbers, it cannot be factored into polynomials with real coefficients because the equation [tex]\( x^2 + 9 = 0 \)[/tex] has no real roots (its roots are [tex]\( \pm 3i \)[/tex]).
- Therefore, [tex]\( x^2 + 9 \)[/tex] is prime.
2. [tex]\( x^2 - 9 \)[/tex]
- This polynomial is [tex]\( x^2 - 9 \)[/tex]. To check if it's prime, we can attempt to factor it. We recognize it as a difference of squares: [tex]\( x^2 - 9 = (x + 3)(x - 3) \)[/tex].
- Since it can be factored into polynomials of lower degree, [tex]\( x^2 - 9 \)[/tex] is not prime.
3. [tex]\( x^2 + 3x + 9 \)[/tex]
- This polynomial is [tex]\( x^2 + 3x + 9 \)[/tex]. To check if it's prime, we need to see if it can be factored into polynomials of lower degree. Factoring this over the real numbers is not straightforward, and we find its discriminant [tex]\( \Delta = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 9 = 9 - 36 = -27 \)[/tex]. A negative discriminant indicates that the polynomial has complex roots and cannot be factored into polynomials with real coefficients.
- Since it cannot be factored into polynomials of lower degree with real coefficients, [tex]\( x^2 + 3x + 9 \)[/tex] is prime.
4. [tex]\( -2x^2 + 8 \)[/tex]
- This polynomial is [tex]\( -2x^2 + 8 \)[/tex]. To check if it's prime, we can attempt to factor it. First, note it can be factored as [tex]\( -2(x^2 - 4) \)[/tex]. Moreover, [tex]\( x^2 - 4 \)[/tex] is a difference of squares: [tex]\( x^2 - 4 = (x + 2)(x - 2) \)[/tex].
- Therefore, [tex]\( -2x^2 + 8 = -2(x + 2)(x - 2) \)[/tex]. Since it can be factored, [tex]\( -2x^2 + 8 \)[/tex] is not prime.
Based on this detailed analysis, the polynomials that are prime are:
- [tex]\( x^2 + 9 \)[/tex]
- [tex]\( x^2 + 3x + 9 \)[/tex]
