How to Divide a Trinomial by a Binomial?

Binomials and trinomials are kinds of Polynomials. Binomial is of degree two and in general is called as a quadratic polynomial, whereas a trinomial is of degree three and so can also be called as a cubic polynomial. A trinomial can be represented as: pX³ + qX² + rX + D. For Example: (40x³ - 72x² + 20x - 4). A binomial is written in same way as we write a Quadratic Equation: Ax² + Bx + C.
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).