Division of Polynomials

Polynomials are expressions involving x raised to a whole number power (exponent). Some examples are:

In this lesson we consider division of polynomials such as:

and .

There are two ways to calculate a division of polynomials. One is long division and a second method is called synthetic division. Synthetic division can be used when the polynomial divisor such as x-2 has the highest power of x as 1 and the coefficient of x is also 1. Otherwise, such as , a long division process must be used. Both methods will be demonstrated.

We will now look at some examples showing how these two procedures are done.

#1. Suppose we are to divide . The long division method is shown below:

We use standard division notation: .

Division of polynomials using this method requires us to divide by x, the first term in the divisor, x-2. We divide this into the first term of getting and then follow the normal procedures for division:

• Step 1. Divide
• Step 2. Multiply
• Step 3. Subtract
• Step 4. Bring down….

After the first division, the problem looks like this:

We have divided x into x³ and the result is x². Multiplying x² by x-2 gives . Subtracting and bringing down the next term give us .

We repeat the procedure dividing x into the first term of giving us –3x. Multiplying, subtracting, and bringing down the next term gives the result shown below.

Following this procedure one more time gives the result below:

This result means: Because we are dividing by x-2 we can do “synthetic division” to get this same result. We note that +2 is subtracted from x in the divisor and set up the problems using only the coefficients of . The initial setup looks like this:

The process is somewhat different than traditional division.

• Step 1. Bring down (the first term is 1).
• Step 2. Multiply with the “divisor” 2 giving 2.
• Step 3. Add giving -3. The result is shown below

After the first step, our problem looks like:

Then we go back to Step 2 and repeat the process: multiply -3 by the “divisor” 2 giving us -6. We add and get -4. The result is shown below on the right.

Continuing this process one more time gives us the result below.

Note that the remainder in this division is 0 and is the last number on the right of the bottom row. We use the bottom row to give coefficients of 1, -3, and -4 for a final answer which is the same result as with long division:

.

#2. Divide: . The long division process is shown below using the standard process for division:

• Step 1. Divide
• Step 2. Multiply
• Step 3. Subtract
• Step 4. Bring down

The results are shown step by step below.

Step 1: Setup the problem:

Step 2: Divide 3x into 3x³, place the result x² above the division line and multiply this by the entire divisor 3x+2. That result is placed below 3x³+5x² and is subtracted. The -7x is the term we bring down to prepare for the next division by 3x.

Step 3: Divide 3x into -3x². The result is shown below including multiplying, subtracting and bringing down the last term, 5.

Step 4: Finish the problem by dividing 3x into -9x, multiplying the result -3 by the “divisor” 3x+2, and doing the final subtraction to get a remainder of 11. The result is shown below.

The division is completed. The final answer involves a remainder of 11. Thus . We cannot use synthetic division because the divisor is 3x+2 and the coefficient of x is not 1.

#3. Synthetic division can be used to calculate because the divisor involves a highest power of x which is 1 and the coefficient of x is 1. The problem is setup below. The first setup shows a 0 which is the coefficient of x² because there is no x² term in the problem. There must be a coefficient for each power of x in order for this method to work.

Step 1 the setup:

Step 2: bring down the 2.

Step 3: Multiply 8 x 4 = 32.  Multiply 2 x 4 = 8 and add.  Add 32 + 4 = 36.

Step 4: The final step is to Multiply 36 x 4 = 144 and add to the – 7.

This final result 137 is the remainder. Therefore, .

#4. Long division is needed to divide  because the coefficient of x is not 1.

Step 1: setup the problem:

Step 2: Divide 2x into 2x⁴ and get x³. Multiply x³ by the divisor 2x+3 and get 2x⁴+3x³. Write this under 2x⁴+x³ and subtract. Bring down the 0x². The result is shown below.

Step 3: Divide 2x into -2x³ and get -x². Multiply -x² by the divisor 2x+3 and get -2x³-3x. Write this under –2x³+x² and subtract. Bring down the 0x. The result is shown below.

Step 4: Divide 2x into 4x² and get 2x. Multiply 2x by the divisor 2x+3 and get 4x²+6x. Write this under 4x²+0x and subtract. Bring down the -2. The result is shown below. There is one more step.

Step 5: Divide 2x into -6x and get -3. Multiply -3 by 2x+3 and get -6x-9. Put  the -6x-9 under the -6x-2 and subtract. The remainder is +7. The result is shown below.

Thus .