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S5 Maths: Remainder and Factor Theorem: Paper 1 (Basics)
S5 Maths: Remainder and Factor Theorem: Paper 1 (Basics)
Ten starters on direct substitution with the remainder and factor theorems.
Direct answers Items 1-3: 6, 19, -1. Items 4-6: yes, yes, a = 5. Items 7-9: 0, x = 1/2, -16. Item 10: 15.
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 | Basics |
02 Questions (10 items)
- FoundationGiven f(x) = x^2 + 3x + 2, find f(1).
- FoundationGiven f(x) = 2x^3 – x + 5, find f(2).
- FoundationFind the remainder when x^3 + 2x^2 – 5x + 1 is divided by x – 1.
- IntermediateIs x – 2 a factor of x^3 – 8?
- IntermediateIs x + 1 a factor of x^2 – 3x – 4?
- IntermediateGiven f(x) = x^3 + ax + 4 and f(1) = 10, find a.
- ChallengeFind the remainder when x^2 – 7x + 10 is divided by x – 5.
- ChallengeWhen using the remainder theorem with divisor 2x – 1, which value is substituted?
- ChallengeGiven f(x) = x^3 – 3x^2 + 4, find the remainder on division by x + 2.
- Error redoA student divides f(x) = x^2 + 2x by x – 3 and answers f(-3) = 3. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 6 | Put x = 1: 1 + 3 + 2 = 6. |
| 2 | 19 | 2 x 8 – 2 + 5 = 16 – 2 + 5 = 19. |
| 3 | -1 | By the remainder theorem put x = 1: 1 + 2 – 5 + 1 = -1. |
| 4 | Yes, because f(2) = 8 – 8 = 0 | Factor theorem: the divisor is zero at x = 2, and f(2) = 0 means it divides exactly. |
| 5 | Yes, because f(-1) = 1 + 3 – 4 = 0 | x + 1 = 0 gives x = -1, and the value is 0, so it is a factor. |
| 6 | a = 5 | 1 + a + 4 = 10, so a = 5. |
| 7 | 0 | Put x = 5: 25 – 35 + 10 = 0, so x – 5 is a factor. |
| 8 | x = 1/2 | Solve 2x – 1 = 0 to get x = 1/2 and substitute it. |
| 9 | -16 | Put x = -2: -8 – 12 + 4 = -16. |
| 10 | Should substitute x = 3; the remainder is 15 | The divisor x – 3 is zero at x = 3, not at x = -3. |
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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