6.6 Dividing Polynomials by Polynomials ; Synthetic
Division

Division of a polynomial by a polynomial like (x^{2} + 2x −15)รท (x + 5) is very
similar to

the long division process we use to divide whole numbers . Let’s review this

procedure first:

• 4356 is called the

dividend

• 34 is called the divisor

• the answer is called the

quotient

Divide, Multiply, Subtract, pull

down.

We write the answer as:

We check the answer by: quotient * divisor + remainder =
dividend

What happens if we get a zero remainder when dividing?

______________________

We follow a similar procedure for division of a polynomial
by a binomial .

x^{2} + 2x −15 is called the **dividend** x + 5 is

called the **divisor **the answer is called the

**quotient**

PROCEDURE:

1.) Divide the first term in the dividend by

the first term in the divisor. This is the first

term in the quotient.

2.) Multiply this by each term in the

divisor.

3.) Subtract.

4.) Bring down

Repeat the process until the remainder

can no longer be divided.

We write the answer as:

We check the answer by making sure that:

quotient * divisor + remainder = dividend

GENERAL PROCEDURE:

1. Write the polynomials in standard form. (in order with
descending

powers ) If there is a missing term, fill it in with a zero times the missing

variable part . (i.e. x^{2} +1 = x^{2} + 0x +1)

2. Divide the first term in the dividend by the first term
in the divisor. This is

the first term in the quotient.

3. Multiply each term in the divisor by the result of the
previous step . Write

the answer beneath the dividend with similar terms under each other.

4. Subtract the product from the dividend .

5. Bring down the next terms and treat this as the new
dividend. Start the

process over again. Repeat this until the remainder can no longer be

divided. This happens when the degree of the remainder is less than

the degree of the divisor.

EXAMPLE: Divide

EXAMPLE: Divide:

EXAMPLE: Divide:

EXAMPLE: Divide:

** SYNTHETIC DIVISION :**

We can use synthetic division to divide polynomials if the divisor is of the
form

x − c. This method is quicker than long division.

Let’s see how the method works with

LONG DIVISION:

SYNTHETIC DIVISION:

EXAMPLE: Use synthetic division to divide