Dividing Polynomials

1Summary 重點摘要

Learn basic polynomial division: divide polynomial by monomial.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Divide by a monomialAt S2 level, the most basic polynomial division is division by a monomial: divide every term of the polynomial by the divisor separately, then add the results. For example, (6x² + 3x) ÷ 3 = 6x²÷3 + 3x÷3 = 2x² + x, and the worked example (15x² - 10x + 5) ÷ 5 is handled term by term to give 3x² - 2x + 1. The idea is really splitting the numerator and cancelling term by term, so the rule need not be memorised blindly. Every term must be divided, including the constant, or the whole answer fails.
2. Step 2: Coefficients and variables separatelyWhen dividing monomials, handle coefficients and variables separately: coefficients divide directly, while variables follow the index law — same base divided means indices subtracted. For example, 8x³ ÷ 2x = (8÷2) × (x³÷x) = 4 × x² = 4x². Likewise x³ ÷ x = x² and -6x ÷ (-2x) = 3, where the variables cancel completely and the quotient is a plain number. If a term's degree falls below the divisor's, as in x ÷ x², negative indices or fractions would appear, which S2 questions normally avoid.
3. Step 3: Be careful with negative signsWith negative signs, remember the sign rules: same signs dividing give positive, opposite signs give negative. For example, (-6x² + 9x) ÷ (-3x): the first term -6x² ÷ (-3x) is negative over negative, giving 2x; the second term 9x ÷ (-3x) is positive over negative, giving -3; combined the answer is 2x - 3. For every term, fix the sign first, then the coefficient, then cancel the variables — three separate actions. Sign errors are the most common cause of lost marks in this lesson, and marking each sign individually guards against them.
4. Step 4: A complete examplePutting the three points together, compute (12x³ - 8x² + 4x) ÷ 4x. First term: 12x³ ÷ 4x has coefficients 12÷4 = 3 and degrees 3-1 = 2, giving 3x²; second term: -8x² ÷ 4x keeps the negative sign and gives -2x; third term: 4x ÷ 4x cancels completely to give 1. Adding them yields 3x² - 2x + 1. Note that the third term becomes the constant 1, not 0 — a frequent beginner's slip, since anything divided by itself equals 1.
5. Step 5: Check by multiplicationDivision can be checked by multiplication: quotient times divisor should equal the dividend. For example, (3x² - 2x + 1) × 4x = 12x³ - 8x² + 4x, exactly the original expression, so the quotient is correct. During the check, multiply term by term and compare every coefficient and sign; any mismatch means the division went wrong somewhere and the work should be re-examined. This check costs only a minute or two yet sharply raises the success rate, so reserving a few minutes for it in an examination is well worthwhile.
6. Step 6: Points to watch in examinationsIn examinations the crucial rule is that every term gets divided, including constants and negative middle terms — missing one can lose the whole question. Write the terms on separate lines, produce each quotient first, and add at the end, keeping the working clear for review; afterwards verify once by multiplication. If a question asks for a value after simplifying, simplify first and then substitute, which makes the arithmetic much lighter; re-check the signs of negative terms, and these small habits secure steady marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Divide by monomial: each term divide separately
• Coefficients divide, exponents subtract
• Be careful with negative signs
• Can't miss terms
• Can check with multiplication