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Mathematics · Year 10 · Path · MA5-POL-P-01

Polynomials

Twelve lessons that build one idea at a time: what a polynomial is, how to operate on it, how to take it apart, and what the pieces say about its graph. Every theorem here is proved from the one before it, so the list is short and the reasons are visible.

12lessons
22syllabus points
8-9hstudy
Start Lesson 1 →
1
What Is a Polynomial
Whole-number powers, and the vocabulary: degree, leading term, constant term, monic
2
Polynomial Notation and Evaluation
P(x) as a name, P(a) as substitution, and equating coefficients
3
Adding, Subtracting and Multiplying
Why a product's degree is predictable and a sum's is not
4
Division and the Four Names
Dividend, divisor, quotient and remainder, then long division with powers of x
5
The Division Identity
Writing a division with no division sign, and substituting into it
6
The Remainder Theorem
Verified against a long division, then proved in a single line
7
The Factor Theorem
A zero remainder means a factor, and where to look for the zero
8
Factorising Completely
The null factor law, cubics and quartics, and knowing when to stop
9
Zeroes, Roots and How Many
Three words for the same numbers, and why the degree caps their count
10
Graphing from Factored Form
Intercepts, end behaviour and sign diagrams: three pieces make a sketch
11
Multiplicity and Shape
Odd multiplicity crosses, even multiplicity touches, higher flattens
12
Transforming Polynomial Graphs
Why a change inside the brackets acts backwards, and which zeroes survive

Focus Area Snapshot

12lessons
22syllabus points
36worked examples
96questions

Where this leads

This is a Path focus area, sitting outside the Core and preparing you for Mathematics Advanced and Extension 1 in Year 11. Polynomials are the single junior topic those courses lean on hardest: the remainder and factor theorems reappear almost unchanged, and every curve-sketching question builds on Lessons 10 to 12.

Start Lesson 1 →