Polynomial Division

1Summary 重點摘要

Learn to divide polynomial by linear expression using long division, find quotient and remainder.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Set out the long division layoutPolynomial long division uses the same layout as numerical long division: the dividend goes inside the division sign, the divisor outside, and both are written in descending order of degree. If a power is missing from the middle of the dividend, insert a term with coefficient 0; for example write x³ - 1 as x³ + 0x² + 0x - 1. Filling in the gaps is the foundation of the whole method: omitting it shifts the terms brought down later and misaligns the subtraction, which is the most common cause of lost marks, so check the terms before starting.
2. Step 2: Divide the leading terms for the first quotient termCompare the leading terms: divide the leading term of the dividend by the leading term of the divisor, and the result is the first term of the quotient. For example x³ divided by x gives x², so the quotient begins with x². The principle is that the next product will remove the leading term of the dividend exactly, so only the leading terms matter in each round. The rule is that coefficients divide while powers subtract, so 2x³ ÷ x = 2x²; beginners often forget to divide the coefficient as well and wrongly write x².
3. Step 3: Multiply back and subtractMultiply the first quotient term by the whole divisor, write the product underneath the dividend aligned with the matching powers, then subtract it from the dividend. For example, when the quotient term is x² and the divisor is x + 1, the product is x³ + x², and both signs flip during the subtraction. The purpose of subtracting is to remove the leading term of the dividend, so the degree of the remainder must drop by one; if it does not, a sign error in the subtraction is almost always the cause, which makes this the best instant checkpoint.
4. Step 4: Bring down the next term and continueAfter subtracting, bring down the next not-yet-used term of the dividend and attach it to the remainder, then divide again by the leading term of the divisor in exactly the same way. The sign must come down together with the term: if the term is -3x, keep the minus sign. This step often goes wrong in a hurry when the constant term is forgotten; cross out each term in the original expression as you bring it down, which guarantees every term is handled exactly once, with nothing repeated or skipped.
5. Step 5: Repeat until the remainder has lower degreeRepeat the four moves of divide, multiply, subtract and bring down until the degree of the remainder is lower than the degree of the divisor, because no new quotient term can then be formed. When the divisor is linear, the remainder is at most a constant; if the final subtraction gives 0, the divisor divides the dividend exactly and the remainder is 0. If the degree ever fails to drop, stop and recheck the previous product and subtraction at once instead of continuing, so that a small slip does not spread through the rest of the working.
6. Step 6: State the quotient and remainder, then verifyAdd up all the quotient terms to form the quotient, and the expression left at the end is the remainder; summarise the result with the identity dividend = divisor × quotient + remainder. For example, with quotient x + 2 and remainder 0 we may write x² + 3x + 2 = (x + 1)(x + 2). After answering, verify with the same identity: multiply the divisor by the quotient and add the remainder, and if the result differs from the dividend, some calculation went wrong, so check the sign of each round's product and subtraction.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Polynomial division like number long division
• Start from highest degree term
• After divide multiply divisor, then subtract
• Remainder degree must be lower than divisor
• Answer: dividend = divisor × quotient + remainder