Division of Polynomials

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 the degree of $A$ is greater than or equal to the degree of $B$ , the process of finding two polynomial $Q$ and $R$ such that

A = B Q + R

and $R$ is of lower degree than $B$ , 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.

Divide $2 x^{3} - 32 x - 15$ by $x - 3$ 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} (2x^{3}-32x-15)-2x^{2}(x-3) & =\cancel{2x^{3}}-32x-15\cancel{-2x^{3}}+6x^{2} & =6x^{2}-32x-15 \end{aligned} or using the long division we have