Division of Polynomials
Let $A=x^{3}+2x^{2}-1$ and $B=x^{2}-x+1$. Then we can write
$ A=BQ+R $where $Q=x+3$ and $R=2x-4$ 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=BQ+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 $2x^{3}-32x-15$ 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{2x^{3}}{x}=2x^{2} $ then multiply $2x^{2}$ 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
Divide $x^{5}-4x^{3}+x^{2}+1$ by $x^{2}+2x-1$ and find the quotient and the remainder
Solution
