Question: Factor the polynomial. \[ 5a^2 - 24a - 5 \] Select the correct choice below and, if necessary,…

Factor the polynomial.

\[ 5a^2 - 24a - 5 \]

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

A. \( 5a^2 - 24a - 5 = \boxed{\phantom{0}} \)

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

B. The polynomial is prime.

Solution

To factor the polynomial \(5a^2 - 24a - 5\), we need to find two numbers whose product is \(5 \times (-5) = -25\) and whose sum is \(-24\). The numbers that satisfy these conditions are \(-25\) and \(1\). Now, rewrite the middle term \(-24a\) using \(-25\) and \(1\): \[ 5a^2 - 25a + a - 5 \] Group the terms: \[ (5a^2 - 25a) + (a - 5) \] Factor out the greatest common factor from each group: \[ 5a(a - 5) + 1(a - 5) \] Now, factor by grouping: \[ (5a + 1)(a - 5) \] Thus, the factored form of the polynomial \(5a^2 - 24a - 5\) is \((5a + 1)(a - 5)\).