Division by Polynomials

Everyone knows mathematical operation of division involving natural numbers. When one number is divided by the other,  we have a number called dividend that is divided by a divisor. We obtain a quotient and a remainder.

Dividend     Divisor  

		quotient
		________________
divisor		Dividend
		-----------      иииииииии          ----------          иииииииии
		remainder

Thus dividend = divisor * quotient + remainder.

The operation of division should be carried on until either the remainder is zero or if the remainder is less than the divisor. If the remainder = 0, then the divisor is a factor of the dividend. On the other hand if the remainder 0, then the divisor is not a factor of the dividend. (Also remember that in case of numbers if divisor >dividend, we get fractional numbers).

In the previous chapter we have seen what are monomials, binomials and polynomials. In this chapter we will see how to divide polynomials.

What we will study in this chapter :

1. How to divide a polynomial  with a monomial
2. How to divide a polynomial with a binomial

1. How to divide a polynomial  with a monomial
Divide x⁴ by x.  
There are various methods of solving this problem.

Method 1 : By factorization :

x⁴   =  x * x * x * x

Now divide this by x

   x * x * x * x  
    =   x * x * x   = x³  
         x  

Method 2 : By using laws of indices 

             x⁴  
                   =  x^{(4 -1)}  = x³  
            x¹  

Method 3 : the usual conventional method of division of numbers.  

		x3
		_______
x		x4
		- x4     _____
		0

In all the three methods, the remainder is 0 and the quotient is x³.

In the method 3, the dividend has to be arranged in the descending order of the indices, then only the division will give a correct answer.

Example 1 :  Divide 6x² + 2x⁴ + 4x + 3   by 2x

If we use the method 3 shown above, we have to first re-arrange the polynomial in the descending power or order of indices.

2x⁴ + 6x² + 4x + 3 2x  

		x3     +     3x    +   2
		______________________________
2x		2x4    +    6x2    +   4x   +   3 -2 x4
		ииииииииииииииииииииии    0     +    6x2   +   4x   +   3                  -6x2                                   иииииииииииииииии                  0     +   4x   +   3                                -4x                             иииииииии                               0   +   3

Step 1 : first all the terms in the dividend are arranged in descending order of the power of indices.

Step 2 : The first term in the quotient is found such that the product of the quotient and the divisor gives zero remainder.

Step 3 : The remaining  terms of the dividend are taken down. Then the procedure from step 1 is repeated.

Step 4 : The division comes to a halt when it is not possible to divide the remaining term of the dividend with the divisor. In this case 3 cannot be divided by 2x, so the division is halted.

Recheck to see if the equation

dividend = divisor * quotient + remainder

holds or otherwise.

Here the dividend is (2x⁴  +  6x²   + 4x+ 3), the divisor is 2x , the quotient is (x³+ 3x + 2) and the remainder is 3.

(2x⁴ + 6x²  + 4x + 3) =  2x  *  (x³+ 3x + 2)  + 3

                               =  (2x⁴ + 6x²  + 4x + 3)  

Example 2 :  Divide   x⁵ + 3x³ ё 2x²    by   x²  

		x3   +   3x  -  2
		____________________
x2		x5   +   3x3 ё 2x2       - x5
		ииииииииииииииииии      0   +  3x3   ё 2x2               - 3x3                                ииииииииии                 0       ё 2x2                        - 2x2                                 +                    иииииии                         0

In this example, one has to be a bit careful while dealing with minus sign. Otherwise the division procedure is as we have learnt before.

2. How to divide a polynomial with a binomial  
Division of a polynomial will become simple when we learn to factorize polynomials. But otherwise the procedure for a division of a polynomial with a binomial is same as discussed earlier. Here again, the dividend needs to be arranged in the descending order of the powers of indices. The  variable with same indices have to be written below each other for obtaining proper subtraction.

Example 1 :  Divide (x² ё 5x  +  6)  by (x  -  3)

		x  - 2
		________________
(x  -  3)		x2ё 5x +  6    - x2  + 3x
		иииииииииииииии           -2x  + 6             -(-2x  + 6)          ииииииииии                   0

The division is complete as the remainder is 0.

Let us cross check our result using the equation

dividend = divisor * quotient + remainder

(x² ё 5x  +  6)   = (x  ё  3) (x ё 2) + 0

                      = x² ё 2x ё 3x + 6   = x² ё 5x  +  6

The quotient (x ё 2) is a factor of the dividend, hence the remainder is zero.

Example  2 : State by actual division if (x² + 2 ) is a factor of (x⁴ + x³  - 3x²  + 3x   -12)

		x2    + x   +  1
		_____________________________
x2 + 2		x4   +  x3  - 3x2  +  3x  - 12     -( x4         + 2x2 )
		иииииииииииииииииииииии      0  +  x3  - x2  +   3x  - 12           -  (x3            +  2x)                иииииииииииииииииииии            0    - x2  +  x   - 12                  - (x2          + 2)                    иииииииииииииии                    0   + x   -  10

The remainder is (x-10) and not 0. Hence the binomial  (x² + 2 )  is not a factor of  
(x⁴ + x³  - 3x²  + 3x   -12).  

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