TopIn our elementary school mathematics we used to divide large Numbers using Long Division method where answer is called quotient and the left over as remainder. To show how is dividing a polynomial by a binomial similar to or different from the long division you learned in elementary school maths let us consider an example.
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.