## Long Division of Polynomials

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 exampleremainder 1 or3 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 ofNote that we must write the numerator and denominator to include all the coefficients ofup to the highest power in the numerator and up to the highest power in the denominator, so that we writeas

At each stage we work to eliminate the highest power ofTo start with the highest power ofis 4: multiply the denominatorby– the first term of the quotient - to getand subtract from the numerator to getNow the highest power ofis 3. We multiply the denominator by– the nest term of the quotient - to getand subtract to getThe highest power ofis 2. We multiply the denominator by 2 – the last term in the quotient - to getand subtract to get zero. HenceThere is no remainder.

If instead we are finding the quotient and remainder ofwe follow the same process, but now, as shown below, the remainder is 9 hence the division is now

hence