Question: Factor the trinomial completely. \[ 5z^4 - 26z^2 - 24 \] Select the correct choice below and,…

Factor the trinomial completely.

\[ 5z^4 - 26z^2 - 24 \]

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

A. \( 5z^4 - 26z^2 - 24 = \) (Factor completely.)

B. The polynomial is prime.

Solution

The problem is to factor the trinomial \(5z^4 - 26z^2 - 24\) completely. First, make a substitution to simplify the problem. Let \(y = z^2\). Then the expression becomes: \[ 5y^2 - 26y - 24 \] Next, factor this quadratic expression. To factor \(5y^2 - 26y - 24\), we look for two numbers that multiply to \(5 \times -24 = -120\) and add to \(-26\). These numbers are \(-30\) and \(4\). Rewrite the middle term using these numbers: \[ 5y^2 - 30y + 4y - 24 \] Now, group the terms: \[ (5y^2 - 30y) + (4y - 24) \] Factor out the greatest common factor from each group: \[ 5y(y - 6) + 4(y - 6) \] Notice that \((y - 6)\) is common: \[ (5y + 4)(y - 6) \] Now, substitute back \(y = z^2\): \[ (5z^2 + 4)(z^2 - 6) \] Thus, the complete factorization of the trinomial \(5z^4 - 26z^2 - 24\) is: \[ (5z^2 + 4)(z^2 - 6) \] The answer is: A. \(5z^4 - 26z^2 - 24 = (5z^2 + 4)(z^2 - 6)\)