Question

select the correct answer.
which expression is a prime polynomial?
a. $8x^{7}-9x + 12x^{2}$
b. $x + 9y^{2}$
c. $x^{9}-y^{6}$
d. $3m + 9n$

Explanation:

Step1: Define prime polynomial

A prime polynomial (irreducible polynomial) cannot be factored into non-constant polynomials with integer coefficients.

Step2: Analyze Option A

Rearrange: $8x^7 + 12x^2 - 9x$. Factor out $x$: $x(8x^6 + 12x - 9)$. It is factorable, so not prime.

Step3: Analyze Option B

$x + 9y^2$ has no common factors, and cannot be factored into non-constant integer-coefficient polynomials. It is prime.

Step4: Analyze Option C

Use difference of squares: $x^9 - y^6 = (x^{\frac{9}{2}})^2 - (y^3)^2 = (x^{\frac{9}{2}} - y^3)(x^{\frac{9}{2}} + y^3)$, or rewrite as $(x^3)^3 - (y^2)^3$ (difference of cubes): $(x^3 - y^2)(x^6 + x^3y^2 + y^4)$. It is factorable, so not prime.

Step5: Analyze Option D

Factor out 3: $3(m + 3n)$. It is factorable, so not prime.

Answer:

B. $x + 9y^2$