To understand how to divide a trinomial by a binomial first we have to factor trinomial and binomial using Factorization method. Suppose we have a division given as: (12x3 - 4x2 - 5x) / (9x2 + 8x + 1). First we will factorize the term (12x3 - 4x2 - 5x) to get its simplest form as: x (12x2 - 4x - 5).
Here, we see that numerator still has a binomial (12x2 - 4x - 5) that can be factorized as: (2x – 1) (6x + 5). In denominator also we need to factorize the binomial (9x2 + 8x + 1) as: (9x + 1) (x + 1).
After factorization our division looks like: (2x – 1) (6x + 5) / (9x + 1) (x + 1). As numerator is not divisible by denominator, we can use technique of partial Fractions as:
(12x2 - 4x - 5) / (9x + 1) (x + 1) = A / (9x + 1) + B / (x + 1),
= (A * (x + 1) + B * (9x + 1)) / (9x + 1) (x + 1),
= (x (A + 9B) + (A + B)) / (9x + 1) (x + 1),
Equating the coefficients we get:
A + 9B = - 4,
A + B = -5,
Solving above two equations we can write, A = 1/8 and B = -11/24,
So, our answer is:
(12x2 - 4x - 5)/ (9x + 1) (x + 1) = (1/8)/ (9x + 1) + (-11/24)/ (x + 1).