1.11 Dividing Polynomials

Read the introductory example

Let $A = x^{3} + 2 x^{2} - 1$ and $B = x^{2} - x + 1$ . Then we can write

$A = B Q + R$

where $Q = x + 3$ and $R = 2 x - 4$ 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 $A$ and $B$ are two polynomials such that $degree (B) \leq degree (A)$ , then there are unique polynomials $Q$ and $R$ such that

$A = B Q + R$

where $degree (R) < degree (B)$ .

The process of finding $Q$ and $R$ is called the process of dividing $A$ by $B$ .
In this process, $A$ is called the dividend, $B$ the divisor, $Q$ the quotient, and $R$ the remainder. If $R = 0$ , we say $A$ is divisible by $B$ .

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

Example

Divide

2 x^{3} - 32 x - 15

x - 3

and find the quotient and the remainder.

Solution

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

x

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

\frac{2 x^{3}}{x} = 2 x^{2}

then multiply

2 x^{2}

by the divisor and subtract the result from the dividend

$\begin{aligned} (2 x^{3} - 32 x - 15) - 2 x^{2} (x - 3) & = 2 x^{3} - 32 x - 15 - 2 x^{3} + 6 x^{2} \\ = 6 x^{2} - 32 x - 15 \end{aligned}$

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 $6 x^{2} - 32 x - 15$ by $x - 3$ and follow the same steps; that is, we write it in descending power of $x$ and divide its first term, $6 x^{2}$ , by the first term of the divisor, $x$ : $6 x^{2} / x = 6 x$

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

Therefore

$2 x^{3} - 32 x - 15 = (x - 3) (2 x^{2} + 6 x - 14) - 57.$

Here the quotient is $Q = 2 x^{2} + 6 x - 14$ and the remainder is $R = - 57.$

Now try to solve the following example.

Example

Divide $x^{5} - 4 x^{3} + x^{2} + 1$ by $x^{2} + 2 x - 1$ and find the quotient and the remainder

Solution

The quotient and the remainder are $Q = x^{3} - 2 x^{2} + x - 3$ and $R = 7 x - 2$ , respectively.

Read the introductory example

Collapse the introductory example