Long division of one number by another, if the divisor is not a factor, results in a decimal number, or a quotient plus a remainder. For example
remainder 1 or
3 is the quotient and 1 the remainder. It is not just pure numbers that can undergo long division. So can polynomials. In the example below it is shown how to find the quotient and remainder of
Note that we must write the numerator and denominator to include all the coefficients of
up to the highest power in the numerator and up to the highest power in the denominator, so that we write
as
At each stage we work to eliminate the highest power of
To start with the highest power of
is 4: multiply the denominator
by
– the first term of the quotient - to get
and subtract from the numerator to get
Now the highest power ofis 3. We multiply the denominator by
– the nest term of the quotient - to get
and subtract to get
The highest power ofis 2. We multiply the denominator by 2 – the last term in the quotient - to get
and subtract to get zero. Hence
There is no remainder.
If instead we are finding the quotient and remainder of
we follow the same process, but now, as shown below, the remainder is 9 hence the division is now
hence