Sketch cubic products and solve their inequalities
Use linear factors to locate cubic intercepts, reflect negative parts for a modulus graph and read inequality intervals.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What are the x-intercepts of (x + 2)(x − 1)(x − 3)?
What you will learn
- Sketch a cubic given as a product of three linear factors
- Label its coordinate-axis intersections
- Solve a cubic inequality from the graph's sign
Roots and end behaviour establish the main sketch
For f(x) = (x + 2)(x − 1)(x − 3), the x-intercepts are −2, 1 and 3. The y-intercept is f(0) = 6. A positive leading coefficient means the graph falls to the left and rises to the right.
Create a sign table using one test value in each interval. The sign changes at each simple root, so this cubic alternates below, above, below and above the axis.
Modulus and inequalities use the same sign information
For |f(x)|, reflect every negative interval above the x-axis; preserve each root. For f(x) ≥ d, compare the graph with the horizontal line y = d and include equality points.
A clear labelled sketch supports the interval answer. Do not use turning-point estimates that the question has not given or that you have not calculated.
Pause and explain
For |f(x)|, what happens where f(x) is below the x-axis?
Worked example
Solve (x + 2)(x − 1)(x − 3) ≥ 0.
Show the worked solution
- The roots are −2, 1 and 3; test the sign in each interval.
- For x = −3 the product is negative; for x = 0 it is positive.
- The sign alternates at each simple root: negative, positive, negative, positive.
Answer −2 ≤ x ≤ 1 or x ≥ 3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the sign of f(x) = (x − 1)(x + 1)(x − 4) on each interval between its roots.
Give me a hint
Use roots −1, 1 and 4, then test one value in each interval.
Compare my reasoning
- At x = −2 the product is negative.
- Each simple root flips the sign.
- The intervals are (−∞,−1): −; (−1,1): +; (1,4): −; (4,∞): +.
Signs: negative, positive, negative, positive across the four intervals.
Look for these in your work
- Roots ordered
- Test point used
- Sign changes tracked
Solve (x − 2)(x + 3)(x − 5) < 0.
Give me a hint
Find the ordered roots, then use the leading coefficient and sign changes.
Compare my reasoning
- Roots are −3, 2 and 5.
- For x > 5 the product is positive; signs alternate crossing each simple root.
- The negative intervals are x < −3 and 2 < x < 5.
x < −3 or 2 < x < 5.
Look for these in your work
- Roots ordered
- Leading sign used
- Strict endpoints excluded
A sensor reports y = |(x − 2)(x + 1)(x − 4)|. State its zeros and explain which parts of the original cubic change on the display.
Give me a hint
Use the cubic's sign intervals before taking its modulus.
Compare my reasoning
- The zeros remain x = −1, 2 and 4.
- The positive intervals remain unchanged.
- The negative intervals reflect upward; the displayed y-values are never negative.
Zeros at −1, 2 and 4; only the below-axis intervals are reflected above the axis.
Look for these in your work
- All intercepts retained
- Sign intervals considered
- Transformation described
Common mistakes
- Omitting the y-intercept when a labelled sketch is required
- Using a sign chart without accounting for every root
- Excluding roots from a non-strict inequality
What can you explain now?
For a positive-leading cubic with three simple roots, what happens to its sign at each root?
Compare with the explanation
It changes sign
A simple linear factor changes sign while the other factors remain non-zero nearby.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Explain how roots, a sign table and end behaviour support a cubic sketch and inequality solution.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.