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High School Mathematics Extensions/Supplementary/Polynomial Division

< High School Mathematics Extensions‎ | Supplementary

IntroductionEdit

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 polynomialsEdit

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

 

Distributing,

 

Finally,

 


Another trickier example making use of factors:

 

Reordering,

 

Factoring,

 

One more time,

 

Yielding,

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

Long divisionEdit

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   .