Question

which polynomials are prime? check all of the boxes that apply.
$x^2 + 9$
$x^2 - 9$
$x^2 + 3x + 9$
$-2x^2 + 8$
done

Explanation:

Step1: Define prime polynomial

A prime polynomial cannot be factored into lower-degree polynomials with integer coefficients.

Step2: Analyze $x^2 + 9$

This is a sum of squares ($x^2 + 3^2$). Sum of squares cannot be factored over integers.

Step3: Analyze $x^2 - 9$

This is a difference of squares. Factor as:
$x^2 - 9 = (x-3)(x+3)$
It is not prime.

Step4: Analyze $x^2 + 3x + 9$

Check discriminant: $\Delta = b^2-4ac = 3^2 - 4(1)(9) = 9-36=-27$. Negative discriminant means no real integer roots, so it cannot be factored over integers.

Step5: Analyze $-2x^2 + 8$

Factor out $-2$ first:
$-2x^2 +8 = -2(x^2-4) = -2(x-2)(x+2)$
It is not prime.

Answer:

$\boldsymbol{x^2 + 9}$, $\boldsymbol{x^2 + 3x + 9}$