Question: Factor the trinomial. \[ t^2 - 8tz + 12z^2 \] Select the correct choice below and fill in any…

Factor the trinomial.

\[ t^2 - 8tz + 12z^2 \]

Select the correct choice below and fill in any answer boxes within your choice.

A. \( t^2 - 8tz + 12z^2 = \boxed{\phantom{0}} \)

B. The polynomial is prime.

Solution

To factor the trinomial \( t^2 - 8tz + 12z^2 \), we can look for two numbers that multiply to \( 12z^2 \) and add up to \( -8tz \). These numbers are \(-6z\) and \(-2z\). Rewrite the middle term \(-8tz\) using \(-6z\) and \(-2z\): \[ t^2 - 6tz - 2tz + 12z^2 \] Group the terms: \[ (t^2 - 6tz) + (-2tz + 12z^2) \] Factor out the greatest common factor from each group: \[ t(t - 6z) - 2z(t - 6z) \] Notice that \(t - 6z\) is a common factor: \[ (t - 6z)(t - 2z) \] Therefore, the factored form of the trinomial is \((t - 6z)(t - 2z)\). Select option A and fill in the box with \((t - 6z)(t - 2z)\).