The prompt is aimed at students who are studying A-level or IB Diploma mathematics. Typically, they will have learnt about graphs of polynomials and basic differentiation before seeing the prompt, although it is possible to introduce both topics during the inquiry.

Students find the prompt intriguing, particularly when they begin to think that it is true for all real values of a, b, and c. As the class verifies more and more individual cases, students begin to ask how it is possible and whether there is a proof of the generalisation.

Question, notice, and wonder

In the question, notice, and wonder phase of the inquiry, students responses to the inquiry have included:

• When the function has three roots, the equation is cubic.

• The graph crosses the x-axis in three places.

• Can two of the roots be equal - repeated roots? 

• We have sketched the graph of the function with roots 1, 2, and 3 and the tangent at 1.5 - the tangent might intercept the x-axis at 3.

• Is the statement true? Does it work for all, some, or no values of a, b, and c.

Generate examples and verify

Using the regulatory cards, students might decide to start the inquiry by creating examples or testing cases. They could choose values of a, b, and c and use the factor theorem to create a cubic equation. When a = 1, b = 2, and c = 3, for example, f(x) = (x - 1)(x - 2)(x - 3) = x3 - 6x2 + 11x - 6.