High School Mathematics Extensions/Supplementary/Polynomial Division

Supplementary Chapters
75% developed Basic Counting
50% developed Polynomial Division
100% developed Partial Fractions
75% developed Summation Sign
75% developed Complex Numbers
75% developed Differentiation
Problems & Projects
0% developed Problem Set
0% developed Exercise Solutions
0% developed Problem Set Solutions



First of all, we need to incorporate some notions about a much more fundamental concept: factoring.

We can factor numbers,


or even expressions involving variables (polynomials),


Factoring is the process of splitting an expression into a product of simpler expressions. It's a technique we'll be using a lot when working with polynomials.

Dividing polynomials


There are some cases where dividing polynomials may come as an easy task to do, for instance:






Another trickier example making use of factors:






One more time,



1. Try dividing   by   .
2. Now, can you factor   ?

Long division


What about a non-divisible polynomials? Like these ones:


Sometimes, we'll have to deal with complex divisions, involving large or non-divisible polynomials. In these cases, we can use the long division method to obtain a quotient, and a remainder:


In this case:

Long division method
1 We first consider the highest-degree terms from both the dividend and divisor, the result is the first term of our quotient.    
2 Then we multiply this by our divisor.  
3 And subtract the result from our dividend.  
4 Now once again with the highest-degree terms of the remaining polynomial, and we got the second term of our quotient.  
5 Multiplying...  
6 Subtracting...  
7 We are left with a constant term - our remainder:  

So finally:

3. Find some   such that   is divisible by   .