Suppose we have two Polynomials given as: (12p

^{3}– 4p

^{2}– 5p) and (9p

^{2}+ 8p + 1). First one is of degree three and is also known as trinomial and one with degree two is called a binomial.

To divide the trinomial by binomial we first factorize degree 3 polynomial to get its simplest form: p (12p

^{2}– 4p - 5) = (2p – 1) (6p +5).

Next we factorize the binomial (9p

^{2}+ 8p + 1) as: (9p + 1) (p + 1). This makes our overall division looks like: (2p – 1) (6p + 5) / (9p + 1) (p + 1). Here we need to use the technique of partial fraction to solve it further as follows:

(12p

^{2 }– 4p - 5) / (9p + 1) (p + 1) = A / (9p + 1) + B / (p + 1),

= (C * (p + 1) + D * (9p + 1)) / (9p + 1) (p + 1),

= (p (C + 9D) + (C + D)) / (9p + 1) (p + 1),

By equating the coefficients of right side of equation to those on left side we get:

C + 9D = - 4,

C + D = -5,

Solving for values of C and D we get final solution as:

(1 / 8) / (9p + 1) + (-11 / 24) / (p + 1),

On dividing the trinomial by binomial using long division we will have same result. Thus we see that dividing a polynomial by a binomial is similar to long division that we studied in our elementary school.