Maths Extension 1 Year 11 - Module 2 - Lesson 13
Polynomial Inequalities
Name
Date
Class
1. Lesson focus
Polynomial inequalities and sign charts. Factor the polynomial, mark all real zeros and use multiplicities to determine where its sign changes.
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. Solve (x−3)(x+1)≤0.
3. Solve x(x−2)²≥0.
3. Challenge
Solve (x+2)(x−1)(x−4)<0.
4. Multiple choice
The solution of (x−4)(x+1)≤0 is what?
A. −1≤x≤4
B. x<−1 or x>4
C. −1<x<4
D. x≤−1 or x≥4
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Solve (x−1)(x+2)>0.
The upward quadratic is positive outside roots −2 and 1: x<−2 or x>1.
2. Solve (x−3)(x+1)≤0.
The product is non-positive between its roots, including endpoints: −1≤x≤3.
3. Solve x(x−2)²≥0.
(x−2)²≥0, so the sign is determined by x; solution x≥0.
Challenge. Solve (x+2)(x−1)(x−4)<0.
Sign alternates at −2,1,4; testing x=0 gives positive, so solution (−∞,−2)∪(1,4).
Multiple choice. The solution of (x−4)(x+1)≤0 is what?
Correct answer: A. The upward quadratic is non-positive between its two roots, with both roots included.
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