Division of Polynomials

Let and . Then we can write

where and are called the quotient and the remainder, respectively. You may verify the above equation by expanding and simplifying the right hand side.

In general, if and are two polynomials such that the degree of is greater than or equal to the degree of , the process of finding two polynomial and such that

and is of lower degree than , is called the process of dividing by . In this process, is called the dividend, the divisor, the quotient, and the remainder. If , we say is divisible by .

How the quotient is obtained is best explained in the following example.

Divide by and find the quotient and the remainder.

Solution

First we make sure that the dividend and the divisor are written in descending powers of

. Next we divide the first term of the dividend by the first term of the divisor

then multiply

by the divisor and subtract the result from the dividend

or using the long division we have

To simplify calculations, we can reverse the signs of the product of the multiplication and then add it to the dividend; namely

Now we divide

and follow the same steps; that is, we write it in descending power of

and divide its first term,

, by the first term of the divisor,

and again repeat until the degree of the remainder becomes less than the degree of the divisor

Therefore

Here the quotient is

and the remainder is

Divide by and find the quotient and the remainder

Solution

The quotient and the remainder are

and

, respectively.

Division of Polynomials

Division of Polynomials

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