Suppose we have two Polynomials given as: (12p3 – 4p2 – 5p) and (9p2 + 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 (12p2 – 4p - 5) = (2p – 1) (6p +5).
Next we factorize the binomial (9p2 + 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:
(12p2 – 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.