Maths Extension 1 Year 11 - Module 2 - Lesson 11
Multiple Roots
Name
Date
Class
1. Lesson focus
Multiple roots and graph behaviour. A root with even multiplicity touches the x-axis without changing sign; a root with odd multiplicity crosses it.
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. Describe the graph behaviour at each root of (x−2)²(x+1).
3. What is the degree and leading coefficient of (x−1)³(x+4)²?
3. Challenge
For (x+3)⁴(x−1), state the multiplicities and end behaviour.
4. Multiple choice
A polynomial root of multiplicity 2 typically does what at the x-axis?
A. has a vertical asymptote
B. crosses with non-zero slope
C. touches and turns
D. cannot be a root
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. For P(x)=(x−2)²(x+1), state each root and multiplicity.
x=2 has multiplicity 2; x=−1 has multiplicity 1.
2. Describe the graph behaviour at each root of (x−2)²(x+1).
It touches/bounces at x=2 (even multiplicity) and crosses at x=−1 (odd multiplicity).
3. What is the degree and leading coefficient of (x−1)³(x+4)²?
Degree=3+2=5; leading coefficient=1.
Challenge. For (x+3)⁴(x−1), state the multiplicities and end behaviour.
Roots −3 (multiplicity 4) and 1 (multiplicity 1); positive leading quintic falls left and rises right.
Multiple choice. A polynomial root of multiplicity 2 typically does what at the x-axis?
Correct answer: C. An even-multiplicity factor does not change sign across the root, so the graph touches and turns.
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