Polynomial Division

The polynomial division is the division between any two polynomials. The division of polynomials can be between a monomial and a polynomial, between two monomials, or between two polynomials. In Mathematics, it helps us to divide two expressions of the algebraic form. Now before going further on how to do polynomial division, we will first understand what polynomials are. Then we will discuss how polynomial division is solved and what are the steps to solve them which we have to remember. We will also discuss some examples so that the topic is clear and you have no doubts.

Concept of Polynomials

In mathematics, the polynomial is the expression that consists of variables and coefficients which involve operations like addition, subtraction, and multiplication of integers. There are basically three types of polynomials: they are monomial, binomial, and trinomial.

The polynomial is written in the form of expression as,

anxn + an-1xn-1 + an-2xn-2 +….……………………. +a2x2 + a1x +a0.

Where n is 0 or a positive variable. In this expression, an an-1, an-2………a2, a1, a0 are the coefficient of the polynomials.

The highest power of the X is known as the degree of the polynomial.

If p(x) represents the expression of the polynomial and we have x = b where p(b) = 0 then b will be the root of the given polynomial.

Examples of polynomials: x4 + 5x3 + 9x, x-5.

Types of Polynomial Division

There are three types of polynomial division which we will discuss in detail.

They are listed below, 

1. Division of one monomial by another monomial.

2. Division of polynomial by monomial.

3. Division of one polynomial by another polynomial.

Division of One Monomial by Another Monomial

In this method, one monomial is divided by the other monomial. There are some simple steps that we have to follow to do the division.

1. First, we will consider two monomials.

2. Then we have to find the factors of the numerator monomial.

3. Similarly, find the factors of the denominator monomial.

4. Now list down the common factors from the numerator and the denominator.

5. Cancel out the common terms from the numerator and the denominator.

6. The remaining terms in the numerator after canceling common terms will be the final answer of the division.

Here we will consider an example so that we understand it in a better way.

Consider the algebraic expression 20X2 divided by 4X.

We will list down the factors of 20X2 = 2 × 2 × 5 × X × X

Factors of 4X = 2 × 2 × X

Now from the numerator and denominator, we will cancel out common terms.

 (2 × 2 × 5 × X × X) / (2 × 2 × X)

Here 2 × 2 × X terms are common in both so they need to cancel out.

Hence 20X2/ 4X = 5X.

Division of Polynomial by a Monomial

In this method, a polynomial is divided by a monomial. We will discuss how a polynomial is divided by a monomial.

While dividing polynomials we have to separately divide each term of the polynomial by the monomial then to get the final result we have to add each term after division.

Let us consider an example, Divide 5x3 + 35xy + 10x by 5x.

We need to separately divide each term of the polynomial by a monomial.

(5x3)/5x + (35xy)/5x + (10x)/5x

= x2 + 7y +2

(5x3 + 35xy+ 10x)/5x = x² + 7y + 2.

Division of a Polynomial by a Polynomial

For dividing two polynomials we have to write them in a standard form that is the dividend and the divisor should be arranged in the decreasing order of their degrees.

This method of division is quite long and complicated. Therefore, the method to divide such polynomials is called ‘long division‘.

In algebra, the method of dividing one polynomial by another polynomial of the same degrees or of the different degrees is called polynomial long division.

Exercise - 

Example 1: Given a polynomial equation, p(x)=x2 – x – 6. Find the zeros of the equation.

Solution:

Given Polynomial, p(x)=x2–x-6

Zeros of the equation is given by,

x2 -3x+2x-6=0

x(x−3)+2(x–3)

(x-3)(x+2) =0

⇒x=3

Or, x = -2

Thus, 3 and -2 are zeros of the given polynomial.