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

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. $$18c^2 - 27cd + 10d^2 = \boxed{\phantom{0}}$$ (Factor completely.)

B. The trinomial is prime.

Solution

Identify the coefficients: \[ a = 18,\ b = -27,\ c = 10 \] Find two numbers \( m \) and \( n \) such that: \[ m \times n = a \times c = 180 & \\ m + n = b = -27 & \] The numbers are: \[ m = -12,\ n = -15 \] Rewrite the middle term using \( m \) and \( n \): \[ 18c^2 - 12cd - 15cd + 10d^2 \] Factor by grouping: \[ 6c(3c - 2d) - 5(3c - 2d) \] Factor out the common binomial: \[ (6c - 5)(3c - 2d) \] Answer: \[ 18c^2 - 27cd + 10d^2 = (6c - 5)(3c - 2d) \]