Question: Factor the trinomial. \[ 41t^2 - 3ty + 11y^2 \] Select the correct choice below and, if…

Factor the trinomial.

\[ 41t^2 - 3ty + 11y^2 \]

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

A. \( 41t^2 - 3ty + 11y^2 = \boxed{\phantom{Factor completely.}} \) (Factor completely.)

B. The polynomial is prime.

Solution

Factor the trinomial: The given expression is: \[ 4t^2 - 3t + 11y^2 \] To factor this expression, we will check if it can be factored as a trinomial of the form \( (at + by)(ct + dy) \). 1. Identify the terms: - The first term is \(4t^2\). - The last term is \(11y^2\). - The middle term is \(-3t\). 2. Since \(4t^2\) and \(11y^2\) have no common factors besides \(1\), try simple product combinations. 3. Check combinations: - Since \(4t^2\) and \(11y^2\) are already products of irreducible numbers, and they don’t multiply simply to give rise to \(-3t\), this trinomial cannot be factored further over the integers. Since no such factorization exists, the polynomial is considered prime. Select option B: The polynomial is prime.