Division of Polynomials

Introduction

Let us recall that Polynomials are algebraic expressions that consist of variables and coefficients.in other words, an algebraic expression in which the variables involved have only non-negative integral powers, is called a polynomial. Let us also recall some useful terms and definitions with regards to polynomials.

Degree of a Polynomial in one variable – In a polynomial in one variable, the highest power of the variable is called its degree.

Degree of a Polynomial in more than one variable – In a polynomial in more than one variables, the sum of the powers of the variable in each term is computed and the highest sum so obtained is called the degree of the polynomial.

Constant polynomial – A polynomial consisting of a constant term only is called is called a constant polynomial. The degree of the constant polynomial is zero.

Greatest Common Divisor – The greatest common divisor of a set is the largest positive integer or polynomial that divides each of the numbers in the set without remainder.

Let us now learn how to divide a polynomial.

Division of Polynomials by Monomials

We will discuss division of polynomials by monomials in two parts – 

1. Division of Monomial by a Monomial
2. Division of Polynomial by a Monomial

Let us discuss these one by one.

Division of Monomial by a Monomial

In maths, we know that to divide a number say, 42 by a number, say 7, it means that we need to determine a number such that when it is multiplied by 6 the product is equal to 42. Clearly, such a number is 7 and we write 42 ÷ 6 = 7. 

The division of a monomial by a monomial is also defined in a similar manner. This means that dividing the monomial, say X, by a monomial say Y, means finding a monomial Z such that X = Y Z which is also written as –

$\frac{x}{y}$ =  Z

Here X is called the dividend, Y is called the divisor and Z is known as the quotient. The following rules are followed when dividing a monomial by a monomial – 

1. The coefficient of the quotient of two monomials is equal to the quotient of their coefficients.
2. The variable part in the quotient of two monomials is equal to the quotient of the variables in the given monomials.

Let us understand the division through an example – 

Example

Divide 12 a ³ b ³ by 3 a ² b

Solution

We need to divide 12 a ³ b ³ by 3 a ² b. we have,

$\frac{12 a^3 b^3}{3 a^2 b}  = \frac{12 x a x a x a x b x b x b}{3 x a x a x b} = 4 x a x b = 4 a b$

Hence, $\frac{12 a^3 b^3}{3 a^2 b}$   = 4 a b

Division of Polynomial by a Monomial

Dividing the polynomial in one variable by a monomial in the same variable will perform the following steps –

1. Obtain the polynomial which is the dividend and the monomial which is the divisor.
2. Arrange the terms of the dividend in descending order of their degrees. For instance, 6 a² + 5a + 8 a³ + 9 will be written as 8 a³ + 6 a² + 5a + 9.
3. Divide each term of the polynomial by the given monomial by using the rules of division of a monomial by a monomial.

Let us understand this one example

Example

Divide 9 m⁵ + 12m⁴ – 6m² by 3m²

Solution

We are required to divide 9 m⁵ + 12m⁴ – 6m² by 3m²

First we will check whether the terms of the given dividend are in descending order of their degrees. We can clearly see that the terms of the given dividend are in descending order of their degrees. We will therefore, move to the next step.

We have,

$\frac{9 m^5+ 12 m^4-6 m^2}{3 m^2} = \frac{9 m^5}{3 m^2} + \frac{12 m^4}{3 m^2} – \frac{6 m^2}{3 m^2}$ = 3 m ³ + 4 m ² – 2

Hence, $\frac{9 m^5+ 12 m^4-6 m^2}{3 m^2}$ = 3 m ³ + 4 m ² – 2

Division of a Polynomial by a Binomial

The division of a polynomial by a binomial is preferably done using the long division method. The following steps are followed for the division of a polynomial by a binomial – 

1. Arrange the terms of the dividend and the divisor in descending order of their degrees.
2. Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.
3. Multiply the divisor by the first term of the quotient and subtract the result obtained from the dividend to obtain the numerator. 
4. Consider the remainder, if any, as a dividend and repeat the step 2 to obtain the second term of the quotient.
5. Repeat the above steps till a reminder is obtained which is either zero or a polynomial of degree less than that of the divisor.

Let us understand it through an example.

Example

Divide 6 + a – 4 a ²  + a ³ by a – 3

Solution

We are required to divide 6 + a – 4 a ² + a ³ by a – 3. We will perform the above steps for this division. 

1. The first step will be to arrange the terms of the dividend and the divisor in descending order of their degrees. We will write the terms of the dividend as well as the divisor in descending order to get, 
   6 + a – 4 a ² + a ³will be written as a ³– 4 a ² +a + 6 and
   a – 3 will be written as a – 3
2. Now, we will divide the first term a³ of the dividend by the first term a of the divisor and obtain $\frac{a^3}{a} = a^2$ as the first term of the quotient.
3. Next, we will multiply the divisor a – 3 by the first term a of the quotient and subtract the result from the dividend a ³– 4 a ² +a + 6. We will get – a ²  + a + 6 as the remainder. 
4. Now, we will take – a ²  + a + 6 as the new dividend and repeat the step 2. We will get the second term as $\frac{- a^2}{a}$ = – a  of the quotient.
5. We will then multiply the divisor a – 3 by the second term, – a of the quotient and subtract the result – a ² + 3 a from the new dividend. We will thus obtain – 2 a  + 6 as the remainder. 
6. Now, we will consider  – 2 a + 6 as the new dividend and divide its first term – 2 a by the first term a of the divisor to get $\frac{- 2 a}{a}$ = – 2 as the third term of the quotient. 
7. We will then multiply the divisor a – 3 and the third term  – 2 of the quotient and subtract the result  – 2 a  + 6 from the new dividend. We will now get 0 as the remainder. 

Thus we can say that 

(6 + a – 4 a ² + a ³ ) ÷ (a – 3 ) = a ² +a – 2 

Division of Polynomials by using Factorisation

We have learnt the long division method for the division of polynomials. Another method for dividing polynomials is by factorising them. We can factorise the dividend and the divisor together and then cancel out the common factors from the numerator and the denominator. The following steps are involved in the division of polynomials using factorisation – 

1. Obtain the polynomial which is the dividend and the monomial which is the divisor.
2. Write the polynomials in the form of p/q
3. Factorize both the denominator and numerator of each of the polynomials. Also, remember to write each expression in standard form. 
4. Reduce the polynomials by cancelling out common factors in the numerator and denominator
5. Rewrite the remaining factors in the numerator and denominator.

Let us understand it through an example.

Example

Divide 3 x³ – 6 x²+3 x by 3 x² – 3 x

Solution

We are required to divide 3 x³ – 6 x²+3 x by 3 x² – 3 x. We shall perform this division by factorising bot the polynomials. We will first write them in the form of p/q to get – 

$\frac{3x^3 – 6x^2+3x}{3x^2 – 3x}$ . Now, in order to divide the polynomials, first we will look for factors that are common to the numerator & denominator. We can see that, 3x is a common factor the numerator & denominator. Hence we will cancel the common factor to get,

$\frac{3x ( x^2-2x+1)}{3x ( x -1 )}$ 

Now, if possible, we will look for other factors that are common to the numerator and denominator. We can see that ( x – 1 ) is also a common factor of the numerator and the denominator as x²-2x+1 can be written as ( x – 1 ) ( x – 1 ) .

Therefore, we now have,

$\frac{(x-1) ( x-1 )}{( x -1 ) }$

= x – 1

Hence, the polynomial 3 x³– 6 x²+3 x when divided by 3 x²– 3 x will give the result x – 1 or

$\frac{3 x^3 – 6 x^2+3x}{3 x^2 – 3x}$ = x – 1

It is important to mention here that the first step for division of polynomials, irrespective of the method of division being used should always be “pulling out” the common factors. Even if this does not factor out the polynomial completely, this will make the rest of the process much easier. 

Polynomial Division Algorithm

We know that if a number is divided by another number, then –

The number that is to be divided is called the Dividend.  

The number by which the dividend is being divided is called the Divisor. 

The result obtained by the process of division is called the Quotient.

The number that is left over after finding the quotient is called the Remainder.

Also,

Dividend = Divisor x Quotient + Remainder

Similarly, if a polynomial is divided by another polynomial, then also 

Dividend = Divisor x Quotient + Remainder

This is generally known as the division algorithm.

Let us verify this through an example.

Example

Divide x ² + 2 x + 3 x ³ + 5 by 1 + 2 x + x ².

Solution

Let us perform the required division by long division method.

The first step will to check whether all the terms of both the polynomials are in the descending order of their degrees.  We have,

x ² + 2 x + 3 x ³ + 5  in the descending order of their degrees will be written as  – 3 x ³ + x ² + 2 x + 5  and 1 + 2 x + x ² is in descending order of their degrees will be written as x ²  + 2 x  + 1

Now, in order to obtain the first term of the quotient, we will divide the highest degree term of the dividend, i.e. 3x³ by the highest degree term of the divisor, i.e. x². We will get – 

$\frac{3 x^3}{x^2}$ = 3 x 

In the next step, in order to obtain the second term of the quotient, we will divide the highest degree term of the new dividend, i.e. –5 x ² by the highest degree term of the divisor, i.e. x ². We will have,

$\frac{- 5 x^2}{x^2}$ = – 5

We will again carry out the division process with –  5 x ² – x + 5  which was the remainder in the previous step to get 9 x + 10. Now we can see that the degree of 9 x + 10 is less than the divisor x ² + 2 x + 1. So, we cannot continue the division any further. Hence, we now have,

( x ² + 2 x + 3 x ³ + 5 ) ÷  ( 1 + 2 x + x ² ) = 3 x – 5 with remainder 9 x  + 10

Let us now verify the polynomial division algorithm for the above division.

In the above example, we have 

Dividend = x ² + 2 x + 3 x ³ + 5

Divisor = 1 + 2 x + x ²

Quotient = 3 x  – 5

Remainder = 9 x + 10

We know that according to the polynomial divisional algorithm,

Dividend = Divisor x Quotient + Remainder ………………. ( 1 )

Substituting the values of the dividend, divisor, quotient and the remainder in the equation ( 1 ) we have,

x ² + 2 x + 3 x ³ + 5 = (1 + 2 x + x ² ) ( 3  x – 5 ) + ( 9 x  + 10 )

Let us now solve the R.H.S of the equation. We will have,

x ² + 2 x + 3 x ³ + 5 = 3 x  (1 + 2 x + x ² ) – 5 (1 + 2 x + x ² ) + ( 9 x  + 10 )

⇒ x ² + 2 x + 3 x ³ + 5 = 3 x  + 6 x ² + 3 x ³ – 5 – 10 x  – 5 x ² + 9 x  + 10

⇒ x ² + 2 x + 3 x ³ + 5 = 3 x ³ + x ² + 2 x  + 5

We can clearly see above the L.H.S = R.H.S

Hence it can be said that the division of polynomials also satisfies the division algorithm. 

Key facts and Summary

1. An algebraic expression in which the variables involved have only non-negative integral powers, is called a polynomial.
2. In a polynomial in one variable, the highest power of the variable is called its degree.
3. In a polynomial in more than one variables, the sum of the powers of the variable in each term is computed and the highest sum so obtained is called the degree of the polynomial.
4. A polynomial consisting of a constant term only is called a constant polynomial. The degree of the constant polynomial is zero.
5. The greatest common divisor of a set is the largest positive integer or polynomial that divides each of the numbers in the set without remainder.
6. When dividing a monomial by a monomial, two rules are observed – a) The coefficient of the quotient of two monomials is equal to the quotient of their coefficients. b) The variable part in the quotient of two monomials is equal to the quotient of the variables in the given monomials.
7. For dividing a polynomial in one variable by a monomial in the same variable, we divide each term of the polynomial by the given monomial by using the division of a monomial by a monomial. 
8. The first step for division of polynomials, irrespective of the method of division being used should always be “pulling out” the common factors. Even if this does not factor out the polynomial completely, this will make the rest of the process much easier. 
9. Division algorithm states that when a polynomial is divided by another then Dividend = Divisor x Quotient + Remainder