Polynomial Long Division
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.
Preparing for Long Division
Before starting the division, there are a few things to prepare:
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Order Terms:
Write the dividend () and the divisor () in descending order of variable powers (from highest power to lowest).
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Complete Terms:
If any term with a specific power is missing (its coefficient is zero), still write that term with a coefficient of 0 as a placeholder. This is crucial for keeping the columns aligned during subtraction.
Example:
If , the term is missing.
So we write it as .
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Set Up Division:
Write the division in long division format, with (the completed form) inside the division symbol and outside.
Steps for Long Division
The division process is performed step-by-step as follows:
- Divide: Divide the first term of by the first term of . Write the result as the first term of the quotient () above the line.
- Multiply: Multiply the quotient term just obtained by the entire divisor .
- Subtract: Write the result of the multiplication below , aligning like terms, then subtract it from to get a temporary remainder.
- Bring Down: Bring down the next term from next to the temporary remainder to form a new polynomial.
- Repeat: Repeat steps 1-4 with this new polynomial until the degree of the temporary remainder is less than the degree of the divisor .
Long Division Example
Divide by .
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Preparation:
- (complete the term)
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Division Process:
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Step-by-Step Explanation:
Iteration 1:
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Divide: Divide the first term by the first term of the divisor :
Write as the first term of the quotient.
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Multiply: Multiply by the divisor :
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Subtract: Subtract the result from the initial polynomial:
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Bring Down: Bring down the next term () to get the new polynomial:
Iteration 2:
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Divide: Divide the first term of the new polynomial by the first term of the divisor :
Write as the next term of the quotient.
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Multiply: Multiply by the divisor :
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Subtract: Subtract the result from the current polynomial:
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Bring Down: Bring down the next term () to get the temporary remainder:
Stop: The degree of the remainder (, degree 1) is less than the degree of the divisor (, degree 2), so the division stops.
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Result:
- The Quotient () is .
- The Remainder () is .
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Writing in Division Algorithm Form:
Based on the division algorithm, we can write the result as:
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Fraction Form:
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Multiplication Form:
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This long division method might seem lengthy, but it is very systematic and reliable for all types of polynomial division.
Exercise
Find the quotient polynomial and the remainder polynomial after dividing by .
State the result in the form .
Answer Key
Complete to become .
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Quotient:
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Remainder:
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Division Algorithm Form: