Maths Extension 1 Year 11 - Module 2 - Lesson 15
Module Synthesis & Exam Technique
Name
Date
Class
1. Lesson focus
Polynomial synthesis and exam technique. Combine factor, remainder and Vieta methods; show exact substitutions and preserve multiplicity when reporting roots.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. Find the remainder when this polynomial is divided by x−2.
3. State the sum and product of the roots.
3. Challenge
Solve x³−2x²−5x+6≤0.
4. Multiple choice
For a monic cubic with roots r₁,r₂,r₃, the constant term equals what?
A. r₁r₂+r₂r₃+r₃r₁
B. r₁+r₂+r₃
C. −r₁r₂r₃
D. r₁r₂r₃
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Factor x³−2x²−5x+6.
It equals (x−1)(x−3)(x+2).
2. Find the remainder when this polynomial is divided by x−2.
P(2)=8−8−10+6=−4, so the remainder is −4.
3. State the sum and product of the roots.
Roots 1,3,−2: sum=2 and product=−6.
Challenge. Solve x³−2x²−5x+6≤0.
Roots −2,1,3; sign alternates across simple roots and leading cubic is negative on the far left. Solution (−∞,−2]∪[1,3].
Multiple choice. For a monic cubic with roots r₁,r₂,r₃, the constant term equals what?
Correct answer: C. Expanding ∏(x−rᵢ) gives constant term (−1)³r₁r₂r₃.
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