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Division Algorithm for Polynomials


In the previous section you saw how to multiply two polynomials. If p(x) = 3x2 - 1 is a polynomial in x and q(x) = 4x + 1 is another polynomial in x, and if the product of these two polynomials is another polynomial, say m (x), then p (x) ´ q (x)= m (x).

 


If r(x) = 0, then polynomial g(x) is a factor of the polynomial p(x).
 

You know that multiplication and division are inverse operations of each other.
 

 Therefore    p(x) = m(x) ¸ q(x)

 or              q(x) = m(x) ¸ p(x) 

 

Example : (3x2 - 1) (4x + 1) = 4x (3x2 - 1) + 1 (3x2 - 1)
                                    = 12x3 - 4x + 3x2 - 1    

                                    = 12x3 + 3x2 - 4x - 1
This implies that
   12x3 + 3x2 -  4x -  1
¸  (4x + 1) = (3x2 - 1)
and
   12x3 + 3x2 -  4x -  1¸
(3x2 - 1) = (4x + 1)
 Let us see how the division process works.

       

Division of a Monomial by a Monomial
Divide the numerals and each like literal in the numerator, by those in the denominator, separately. 

 

Example

(i) Divide 18a2b by 6ab.

Solution

 

Example

Divide -21 pq2r by -7pqr

Solution

 


Division of a Polynomial by a Monomial
Each term of the polynomial is divided by the monomial.

 

Example

Divide (–5x3y + 3x2 – 6x) by 10x

Solution

 

Example
  Divide
Solution


Second method: In this method we factorise the polynomial in such a way that one of its factors is the given monomial. Expressing the given polynomial as product of one of the monomial and dividing it by the given monomial gives the result.
 

Example

Divide (5y4 + 15y2 - 3y) by 5y

Solution

We see that, 5y4 + 15y2 - 3y
Hence,

Division of a Polynomial by a Binomial

Example

Divide (3x2 + 12x3 + 4x + 1) by (1 + 4x)

Solution

Step 1
Arrange divisor and dividend in decreasing order of the power of the variable.
            

Step 2
Divide the first term of the dividend (12x3) by the first term of the divisor (4x) to get the first term of the quotient (12x3  ¸ 4x =  3x2)

        

Step 3
Proceed as with long division of whole numbers
 
        12x3 + 3x2                ----> 3x2  x (4x +1)
       ____________________
                   0 + 4x + 1    -----> Subtract: bring down all other terms
                         4x + 1   -----> 4x ¸ 4x = 1 gives 2nd term of quotient
                __________________  1 x (4x + 1) = 4x + 1
                              0     -----> subtract; no remainder                                                
Quotient = 3x2 + 1,  Remainder = 0

Check : (Quotient ´ Divisor) + Remainder = Dividend
(3x2 + 1)(4x +1) + 0 = 12 x3 + 3x2 + 4x + 1= Dividend; so the answer is correct.

Second method

Here we factorise the polynomial in such a way that one of its factors is the given binomial. Cancelling out the common factor we arrive at the result.
 

Example

Divide (x3 + 8) by (x + 2)

Solution
(x3 + 8)/(x + 2) = (x + 2)(x2 - 2x + 4)/(x + 2) = (x2 - 2x + 4)    

 

Division of a Polynomial by a Polynomial
Division of a polynomial by a polynomial will result in another polynomial, only if the degree of the divisor is the same as or less than the dividend. The division process is the same as that of division with whole numbers.

The points to be kept in mind are as follows.

 

(i) The terms of both the dividend and divisor should first be arranged in descending order of the powers of the variable.

 

(ii) To start the division, first divide the first term of the dividend with the first term of the divisor to get the first term of the quotient.

 

(iii) Then proceed, as you would do with whole number till you obtain either 0 or a polynomial of degree less than the divisor, as the remainder.
 


 

 

Example

Divide (x4 - 3x2 + 4x + 5) by( x2 - x + 1)

Solution

Arrange the polynomials in descending order of its degree and if the degree of the polynomial is missing write its coefficient as zero.



Hence when we divide

(x4 - 3x2 + 4x + 5) by (x2 - x + 1), we get (x2 + x - 3) as the quotient and 8 as the remainder.






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