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S5 Maths: Remainder and Factor Theorem: Paper 4 (Error clinic)
S5 Maths: Remainder and Factor Theorem: Paper 4 (Error clinic)
Ten items on sign slips, forgetting a non-zero remainder and misusing substitution for quadratic divisors.
Direct answers Items 1-3: 0, 0, 0. Items 4-6: 2, no (5), k = 1. Items 7-9: a = 2; 7 and -1; 0. Item 10: x + 1.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S5 Maths: Remainder and Factor Theorem |
| Level | Error clinic |
02 Questions (10 items)
- FoundationFind the remainder when x^3 + 2x^2 – x – 2 is divided by x + 2.
- FoundationFind the remainder when x^2 + 6x + 5 is divided by x + 5.
- FoundationFind the remainder when 2x^2 – 7x + 5 is divided by 2x – 5.
- IntermediateFind the remainder when x^3 – x + 2 is divided by x + 1.
- IntermediateIs x – 1 a factor of x^3 + 4?
- Intermediatex^2 + kx – 6 is divisible by x – 2. Find k.
- ChallengeThe remainder of x^3 + ax^2 – 4 divided by x – 1 is -1. Find a.
- ChallengeThe remainder of f(x) divided by (x – 3)(x + 1) is 2x + 1. Find f(3) and f(-1).
- ChallengeFind the remainder when 4x^2 + 12x + 9 is divided by 2x + 3.
- Error redoA student divides x^3 + 1 by x^2 – 1 and answers f(1) = 2. What is the correct remainder?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 0 | Put x = -2: -8 + 8 + 2 – 2 = 0. |
| 2 | 0 | Put x = -5: 25 – 30 + 5 = 0, so x + 5 is a factor. |
| 3 | 0 | Put x = 5/2: 12.5 – 17.5 + 5 = 0. |
| 4 | 2 | Put x = -1: -1 + 1 + 2 = 2. |
| 5 | No, the remainder is 5 | f(1) = 1 + 4 = 5, which is not zero, so it is not a factor. |
| 6 | k = 1 | f(2) = 4 + 2k – 6 = 0, so k = 1. |
| 7 | a = 2 | 1 + a – 4 = -1, so a = 2. |
| 8 | f(3) = 7, f(-1) = -1 | Substituting at the roots: 2(3) + 1 = 7 and 2(-1) + 1 = -1. |
| 9 | 0 | Put x = -3/2: 9 – 18 + 9 = 0. Using x = -3 wrongly gives 9. |
| 10 | x + 1 | With a quadratic divisor the remainder is linear: x^3 + 1 = x(x^2 – 1) + x + 1. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Set the divisor to zero, substitute that x into the polynomial and compare with zero.Tutor Registration →
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