Question: Factor the binomial completely. Select “Prime” if the polynomial cannot be factored. \[ 25p^2q^2…

Factor the binomial completely. Select “Prime” if the polynomial cannot be factored.

\[ 25p^2q^2 - 4 = \boxed{\phantom{0}} \]

Prime

Solution

To factor the binomial \( 25p^2q^2 - 4 \), we can recognize it as a difference of squares. The difference of squares formula is: \[ a^2 - b^2 = (a-b)(a+b) \] We identify: \( a^2 = 25p^2q^2 \) \( b^2 = 4 \) First, find \( a \) and \( b \): \[ a = \sqrt{25p^2q^2} = 5pq \] \[ b = \sqrt{4} = 2 \] Now, apply the difference of squares formula: \[ 25p^2q^2 - 4 = (5pq - 2)(5pq + 2) \] Thus, the binomial factors completely as: \[ (5pq - 2)(5pq + 2) \]