Long Division of Polynomials and Synthetic Division

Long Division of Polynomials is very similar to long division with simple numbers.  If you need a refresher on long division watch this video before continuing. 
 http://www.youtube.com/watch?v=8OpsFZqH2ew



Long division allows us to solve for how many times X+2 goes into this function.

Notice that the process is the same as normal long division. 

 

Note that the remainder is -5, but DO NOT write the remainder.  Instead, add to your answer whatever was left (in this case -5) divided by the divisor.  This gives the answer of

Let's look at a different problem.  Notice that the function has no terms for x to the 4th, 3rd or 1st power.

These terms that don't appear to be part of the original function are not skipped.  They are added to the function while being multiplied by zero because they still need to be used to solve the problem even if they cannot be seen in the original function.
 
An additional explanation for Long Division of Polynomials
http://www.youtube.com/watch?v=4u8_AMacu-Y


Long Division is rather tedious.  Fortunately, there is a much faster approach called Synthetic Division. 

Let's look at the original example problem again, but this time use synthetic division. 

 How exactly did that work?


 Write the different values in front of x here

 The first and second line are added here to get a new number, which is then multiplied by the -2



(X+2) was the original function, but you want to take the opposite because (X+2)=0 simplifies to X=2
Zero is included in the top row for each missing term in the dividend.  Step one is to drop down the one because it is the first number in the box.  The bottom row is multiplied by the -2 throughout the problem and the middle row is used for addition with the top row.
The 1 is multiplied by -2 and then is placed below the 7 where it is added to -7, yielding -5.  The product is then put in the next row next to the -2 and the process continues until the box is filled.  The bottom right hand corner (in this case -5) is the remainder.

Well what do the numbers 1, 5, 4, -5 do?
 They are the exact same answer as when this problem was solved using long division.

 We can see them all fit into the answer and the remainder is divided by the divisor.  Be careful about mixing up the signs on the remainder.  We used -2 for synthetic division, but the quotient is still (X+2).

 Helpful video for Synthetic Division
http://www.youtube.com/watch?v=bZoMz1Cy1T4

Adam Peirce