Question: Factor the polynomial. \[ p^2 - q^2 - 4 \] Select the correct choice below and, if necessary,…

Factor the polynomial.

\[ p^2 - q^2 - 4 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \( p^2 - q^2 - 4 = \boxed{\phantom{0}} \)

(Type your answer in factored form. Simplify your answer.)

B. The polynomial is prime.

Solution

To factor the polynomial \( p^2 - q^2 - 4 \), we proceed as follows. First, rewrite the polynomial: \[ p^2 - q^2 - 4 \] Next, observe that \( p^2 - q^2 \) is a difference of squares, which can be factored: \[ p^2 - q^2 = (p - q)(p + q) & \] Now, the expression becomes: \[ (p - q)(p + q) - 4 \] To factor further, we would need to combine or manipulate the terms to find common factors. However, the constant term \(-4\) does not allow for a straightforward factorization with the existing terms. Since we cannot factor the polynomial any further using real numbers, the polynomial \( p^2 - q^2 - 4 \) is prime. Answer: B. The polynomial is prime.