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Polynomial long division is the most common method used to find the quotient and remainder when dividing two polynomials. This method is similar to the long division we perform with integers.
Before starting the division, there are a few things to prepare:
Order Terms:
Write the dividend () and the divisor () in descending order of variable powers (from highest power to lowest).
Complete Terms:
If any term with a specific power is missing (its coefficient is zero), still write that term with a coefficient of as a placeholder. This is crucial for keeping the columns aligned during subtraction.
Example:
If , the term is missing.
So we write it as .
Set Up Division:
Write the division in long division format, with (the completed form) inside the division symbol and outside.
The division process is performed step-by-step as follows:
Divide by .
Preparation:
Division Process:
Step-by-Step Explanation:
Iteration :
Divide: Divide the first term by the first term of the divisor :
Result:
Writing in Division Algorithm Form:
Based on the division algorithm, we can write the result as:
Fraction Form:
This long division method might seem lengthy, but it is very systematic and reliable for all types of polynomial division.
Find the quotient polynomial and the remainder polynomial after dividing by .
State the result in the form .
Complete to become .
Quotient:
Remainder:
Division Algorithm Form:
Write as the first term of the quotient.
Multiply: Multiply by the divisor :
Subtract: Subtract the result from the initial polynomial:
Bring Down: Bring down the next term () to get the new polynomial:
Iteration :
Divide: Divide the first term of the new polynomial by the first term of the divisor :
Write as the next term of the quotient.
Multiply: Multiply by the divisor :
Subtract: Subtract the result from the current polynomial:
Bring Down: Bring down the next term () to get the temporary remainder:
Stop: The degree of the remainder (, degree ) is less than the degree of the divisor (, degree ), so the division stops.
Multiplication Form:
