Level 5 · Mathematics M5.1

Functions, polynomials and the binomial

Work with polynomials, use the factor theorem, and expand brackets raised to any power.

Planned. This book is not written yet. Below is its plan, which a teacher can already use.

You are ready for this book if

You can factorise x² − x − 6 and find f(2) when f(x) = x³ − 4.

This book M5.1 Functions, polynomials and the binomial
One step down M4.10 Functions and graphs

The plan

M5.1.1 Domain, range and inverses
- Find the domain and range of a function
- Say when a function has an inverse
- Sketch a function and its inverse, mirrored in y = x
M5.1.2 The modulus function
- Sketch y = |2x − 3|
- Solve |x − 2| < 5
M5.1.3 Dividing polynomials
- Divide x³ + 2x² − 5x − 6 by x + 1
- Find a remainder quickly with the remainder theorem
M5.1.4 The factor theorem
- Use f(a) = 0 to show that (x − a) is a factor
- Factorise a cubic fully and sketch its graph
M5.1.5 Partial fractions
- Split (3x + 5)/((x + 1)(x + 2)) into two simpler fractions
- Split a fraction with a repeated factor on the bottom
M5.1.6 Pascal's triangle and choosing
- Build Pascal's triangle and link it to (a + b)ⁿ
- Find nCr, the number of ways to choose r things from n
M5.1.7 The binomial expansion
- Expand (2 + x)⁵ with the binomial theorem
- Find one term, like the x³ term, without the full expansion
- Expand (1 + x)⁻¹ for small x and say when it is valid

For parents and teachers

Level 5 fits learners who can already: read a textbook. use functions and graphs. In many schools that is about 17 to 20, but start wherever the learner is.

Strand: Algebra. Written and drawn by attnom. Free to read, print and share.