Question: Factor the trinomial completely. \[ 5z^4 - 7z^2 - 24 \] Select the correct choice below and, if…

Factor the trinomial completely.

\[ 5z^4 - 7z^2 - 24 \]

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

A. \( 5z^4 - 7z^2 - 24 = \boxed{\text{(Factor completely.)}} \)

B. The polynomial is prime.

Solution

To factor the trinomial \(5z^4 - 7z^2 - 24\) completely: Let \(y = z^2\). Then the expression becomes: \[ 5y^2 - 7y - 24 \] We will factor by finding two numbers whose product is \(5 \times (-24) = -120\) and whose sum is \(-7\). The numbers are \(8\) and \(-15\). Rewrite the middle term: \[ 5y^2 + 8y - 15y - 24 \] Group the terms: \[ (5y^2 + 8y) + (-15y - 24) \] Factor each group: \[ y(5y + 8) - 3(5y + 8) \] Factor out the common factor \((5y + 8)\): \[ (y - 3)(5y + 8) \] Substitute back \(y = z^2\): \[ (z^2 - 3)(5z^2 + 8) \] The completely factored form is: \[ (z^2 - 3)(5z^2 + 8) \]