## 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 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 of is 3. We multiply the denominator by – the nest term of the quotient - to get and subtract to get The highest power of is 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  