Polynomial Division & Synthetic Division

• Long Division of Polynomials

So far we have looked at addition, subtraction, and multiplication of polynomials. One very important topic is division of polynomials. There are many uses for division. First, we’ll start off with the most basic form of division, Long Division. It is similar to regular long division of numbers.

• Basics:

Divide the Polynomial:
f(x)= 2x² + 10x+12 by (x+3)

1. First, we look at the first term of the Dividend, 2x²:
   and the first term in the divisor x+3
   
2. Now ask yourself, what can the “x” in x+3 be multiplied with in order to get 2x²?
   What we can do is multiply by 2x.
   After that multiply all of x+3 by 2x and put the values you are going along.
   
3. Next subtract what is left over:
   
4. Now its the same question with finding the factor. Do the same process with 4x+12.
   You will see that you should try +4. Then you will get 4x+12 which is the same thing.
   Simply subtract and you will get zero.
   This tells you that there is no remainder.
   Therefore your quotient is: 2x +4
   
   

• Remainder Problem with long division:

Divide 6x³+10x²+x+8 by (2x²+1)

1. First look for that first term again. 3x multiplied by 2x² will give you the 6x³.
   
2. Multiply the rest and subtract:
   When your multiplying, you will see there is no “x²" term, therefore, you put 0 in place for a placeholder.
   
3. You will now wind up with another polynomials. Repeat the process again. 5 will work in this case.
   
4. Now subtract with what’s left over.
   
5. Finally we come up with (-2x-3)
   Is there any term which will give us (-2x)? No. Therefore this is the remainder. It is written as:
   

Synthetic Division:

Synthetic Division is a very simple way of dividing polynomials . The only thing to keep in mind is that synthetic division only works for the form x-k. You cannot use it divide polynomials by a quadratic such as x²-6.

Here is the general pattern of division:

As you can see first you drop the first term. Then you simply muliply diagonally and then add or subtract vertically.

Here is an example:

Divide: x⁴ -10x² -2x+4 by (x+3)

And now here are some theorems to help you understand some basic concepts:

The Division Algorithm States:

f (x) = d(x)q(x) + r (x)

If the remainder r(x)=0 then d(x) divides evenly into f(x).

Some things to keep in mind are: d(x) cannot =0 and the degree of d(x) is less than or equal to the degree of f(x).

The Remainder theorem:

If a polynomial f(x) is divided by x-k , then the remainder is r = f(k).

This basically states that if you substituted k for the value of x when evaluating the function f(x), you would get the remainder.

The Factor Theorem:

A polynomial f(x) has a factor (x- k) if and only if f(k) = 0.

This means that a factor can only be a factor if f(k) comes out to zero which means that it divides evenly into the function.

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