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: Recall Prime Polynomial

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

Step2: Analyze Option A

For the polynomial \(8x^{7}-9x + 12x^{2}=x(8x^{6}+12x - 9)\). Since we can factor out an \(x\) (a non - constant factor), it is not a prime polynomial.

Step3: Analyze Option B

The polynomial \(x + 9y^{2}\). We try to factor it. There are no two binomials of the form \((ay + b)(cy + d)\) (or other forms) with integer coefficients that multiply to \(x+9y^{2}\). Also, it has no common factors (the terms have different variables and no common numerical factors other than 1). So it is a prime polynomial.

Step4: Analyze Option C

Using the difference of cubes formula \(a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})\) and the fact that \(x^{9}=(x^{3})^{3}\) and \(y^{6}=(y^{2})^{3}\). So \(x^{9}-y^{6}=(x^{3})^{3}-(y^{2})^{3}=(x^{3}-y^{2})(x^{6}+x^{3}y^{2}+y^{4})\). It can be factored, so it is not a prime polynomial.

Step5: Analyze Option D

For the polynomial \(3m + 9n=3(m + 3n)\). We can factor out a 3 (a non - constant factor), so it is not a prime polynomial.

Answer:

B. \(x + 9y^{2}\)