Question: Factor the trinomial. \[ 18c^2 - 27cd + 10d^2 \] Select the correct choice below and, if…

Factor the trinomial.

\[ 18c^2 - 27cd + 10d^2 \]

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

A. \( 18c^2 - 27cd + 10d^2 = \) (Factor completely.)

B. The trinomial is prime.

Solution

To factor the trinomial \(18c^2 - 27cd + 10d^2\), we need to check if it can be factored into two binomials. First, look for a pattern or common factors. There are no common factors for all the terms. Next, use the factoring method for quadratic trinomials. To factor, find two numbers that multiply to \(18 \times 10 = 180\) and add up to \(-27\). Those numbers are \(-15\) and \(-12\). Rewrite the middle term using these numbers: \[ 18c^2 - 15cd - 12cd + 10d^2 \] Group the terms: \[ (18c^2 - 15cd) + (-12cd + 10d^2) \] Factor out the greatest common factor from each group: \[ 3c(6c - 5d) - 2d(6c - 5d) \] Notice that \(6c - 5d\) is a common factor: \[ (3c - 2d)(6c - 5d) \] Therefore, the factored form of the trinomial is: \[ (3c - 2d)(6c - 5d) \]