M5.1.4 For the adult
The factor theorem
How can one number tell me that x − 2 divides a polynomial exactly?
What this chapter teaches
The factor theorem says x − a is a factor of f(x) exactly when f(a) = 0. To factorise a cubic, try the divisors of the constant term until one gives 0, divide that factor out, and factorise the quadratic that is left. The roots show where the graph crosses the x axis, and the constant term is where it meets the y axis.
Three questions to ask
There is no one right wording. Listen for the idea.
- If f(3) = 0, which bracket is a factor of f(x)?
- Which numbers would you try first for a root of x³ − 3x² − 4x + 12, and why?
- What does the graph do at a repeated root such as the 1 in (x − 1)²(x + 2)?
After the chapter, the learner can
- Use f(a) = 0 to show that (x − a) is a factor
- Factorise a cubic fully and sketch its graph
What they copy into their notebook
f(a) = 0 means x − a is a factor. Try divisors of the constant term, divide, then factorise the quadratic. Roots are crossings. A repeated root touches.