Question: Factor the polynomial. \[ p^2 - q^2 - 25 \] Select the correct choice below and, if necessary,…

Factor the polynomial.

\[ p^2 - q^2 - 25 \]

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

A. \( p^2 - q^2 - 25 = \boxed{\phantom{0}} \)

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

B. The polynomial is prime.

Solution

To factor the polynomial \( p^2 - q^2 - 25 \), we’ll first see if it can be expressed as a difference of squares. Notice that the expression can be rearranged to: \[ p^2 - (q^2 + 25) \] Now, check if \( q^2 + 25 \) can be expressed as a square, but it cannot because 25 is not a square number when combined with \( q^2 \). Thus, the polynomial cannot be factored further into a product of binomials with real coefficients. Since \( q^2 + 25 \) cannot be factored further, the polynomial is not easily factorable in real numbers. Therefore, the polynomial is prime. The correct choice is: B. The polynomial is prime.