Question: Factor the trinomial. \[ 75t^3 - 30t^2 + 3t \] Select the correct choice below and, if…

Factor the trinomial.

\[ 75t^3 - 30t^2 + 3t \]

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

A. \( 75t^3 - 30t^2 + 3t = \boxed{\phantom{Factor completely.}} \) (Factor completely.)

B. The polynomial is prime.

Solution

Factor the trinomial: First, recognize the expression to factor: \[ 75t^3 - 30t^2 + 3t \] Identify the greatest common factor (GCF) of the terms. The GCF is \(3t\). Factor out the GCF: \[ 3t(25t^2 - 10t + 1) \] Next, check if the quadratic \(25t^2 - 10t + 1\) can be factored further. Identify factors of \(25 \times 1 = 25\) that add to \(-10\): The factors are \(-5\) and \(-5\). Rewrite the quadratic: \[ 25t^2 - 5t - 5t + 1 \] Group the terms: \[ (25t^2 - 5t) - (5t - 1) \] Factor by grouping: \[ 5t(5t - 1) - 1(5t - 1) \] Combine the factors: \[ (5t - 1)(5t - 1) \] The completely factored form is: \[ 3t(5t - 1)^2 \]