
Long Division
To divide a polynomial , we make use of the long division process. Before bginning the long division process, write the dividend and the divisor in descending powers of x. In order to help align like terms, use 0 as the coefficient of any missing power.
Example 1: Divide
(2x^{3} + 3x^{2}  x + 16) / (x^{2} + 2x  3)
Solution:
The dividend 2x^{3} + 3x^{2}  x + 16 is a polynomial of degree 3. The divisor x^{2} + 2x  3 is a polynomial of degree 2. By laws of the exponents, we expect that the quotient is a polynomial of degree 1.
STEP 1: We first divide the term with highest power in the dividend 2x^{3} by the term with the highest power in the divisor x^{2} to obtain a quotient equal to 2x and organize all three terms as follows.
 2x  
 ____________________ 
x^{2} + 2x  3   2x^{3} + 3x^{2}  x + 16  
STEP 2: We next multiply the divisor x^{2} + 2x  3 by the quotient 2x and organize the result as follows
 2x  
 ____________________ 
x^{2}+2x3  2x^{3} + 3x^{2}  x + 16  
 2x^{3} + 4x^{2}  6x   Multiply x^{2} + 2x  3 by 2x 
STEP 3: We next subtract the result of the multiplication from the dividend as follows
 2x  
 ____________________ 
x^{2}+2x3  2x^{3} + 3x^{2}  x + 16  
 2x^{3} + 4x^{2}  6x   
 _______________________   
 x^{2} + 5x + 16   Subtract 
STEP 4: We now divide the term with the highest power in the subtraction result x^{2} by the term with the highest power in the divisor x^{2} to obtain 1 and organize all terms as follows
 2x  1  
 ____________________ 
x^{2}+2x3  2x^{3} + 3x^{2}  x + 16  
 2x^{3} + 4x^{2}  6x   
 _______________________   
 x^{2} + 5x + 16   
STEP 5: We next multiply the divisor x^{2} + 2x  3 by 1 and organize all terms as follows
 2x  1  
 ____________________ 
x^{2}+2x3  2x^{3} + 3x^{2}  x + 16  
 2x^{3} + 4x^{2}  6x   
 _______________________   
 x^{2} + 5x + 16   
 x^{2} 2x + 3   Multiply 2 + 2x  3 by 1 
STEP 6: Subtract the result of the last multiplication from the term before it and organize the results as follows.
 2x  1  
 ____________________ 
x^{2}+2x3  2x^{3} + 3x^{2}  x + 16  
 2x^{3} + 4x^{2}  6x   
 _______________________   
 x^{2} + 5x + 16   
 x^{2} 2x + 3   
 _______________________   
 7x + 13   Subtract 
We now stop the process since the last term 7x + 13 has a degree smaller that that of the divisor x^{ 2} + 2x  3.
The result of the long division may be written as follows
(2x^{3} + 3x^{2}  x + 16) / (x^{2} + 2x  3) = (2x  1) + (7x + 13) / (x^{2} + 2x  3)
or also as follows
2x^{3} + 3x^{2}  x + 16 = (2x  1)(x^{2} + 2x  3) + (7x + 13)
Vocabulary associated with the long division process
2x^{3} + 3x^{2}  x + 16 is the dividend
x^{2} + 2x  3 is the divisor
2x  1 is the quotient
7x + 13 is the remainder
More References and Links to Polynomials
polynomial Functions.
