Vu's Maths Hub: HSC Extension 1 Polynomials: Factor Theorem, Division, and Graphs

Intro

The HSC Polys Ext 1 booklet is a free Mathematics Extension 1 resource with 45 worked problems covering polynomial language, remainder and factor theorems, division algorithms, sums and products of zeroes, and graph sketching. It is written for Extension 1 students wanting focused exam-oriented polynomial practice and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Polys Ext 1 — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Polys Ext 1 booklet offers a fundamentals review, 45 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Polys Ext 1 — no account required. Use this post to plan how to work through a 70-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 1 students wanting focused exam-oriented polynomial practice.

Focus: polynomial language, remainder and factor theorems, division algorithms, sums and products of zeroes, and graph sketching.

Topics covered:

Polynomial language and notation
Graphs of polynomial functions
Division algorithms
Remainder and factor theorems
Sums and products of zeroes
Multiple zeroes
Long and synthetic division
Graph sketching

How to use it:

Read core formulas first; attempt before solutions
Classify: factoring, root extraction, or proof
Look for symmetry and repeated factors
Use remarks and takeaways after each solution

Approximately 70 pages, CC BY 4.0, readable at HSC Polys Ext 1.

Key fundamental reviews

Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:

Polynomial notation and degree
Long division and synthetic division
Remainder and factor theorems
Sums and products of zeroes (Vieta-style at Ext 1 level)
Multiplicity and turning points on graphs
Sketching from factored form

Why fundamentals matter

Polynomial notation and degree — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Long division and synthetic division — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Remainder and factor theorems — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Sums and products of zeroes — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Multiplicity and turning points on graphs — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Polys Ext 1 fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Polys Ext 1 booklet packs 45 practice problems into roughly 70 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Part 1 — detailed solutions: basic, medium, and advanced polynomial problems with full solutions.

Part 2 — hint-based fluency: warm-up drills, stretch problems, and challenge corner.

Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.

Part 1 (35 problems)

Advanced (14 problems)

This tier contains 14 problems aimed at extension and synthesis. Representative work includes "A Similar Sum-of-Squares Quartic" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

A Similar Sum-of-Squares Quartic
Intersections of (x+1)^n and (x+1)^m
A Factor-Theorem Identity in Three Variables
A Common Quadratic Factor
Intersections via P(x)-Q(x)
The Reversal Transformation
The Cosine-Polynomial Bridge
The Tangent Cubic
The Fibonacci Polynomial Remainder
The Chebyshev Recurrence
Hermite and the Bell Curve
Bernoulli Polynomials and the Beta Distribution
Legendre Polynomials and the Infinite Dot Product
Telescoping Polynomials and Trigonometry

Basic (8 problems)

This tier contains 8 problems aimed at foundational fluency. Representative work includes "Identically Equal Polynomials" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Identically Equal Polynomials
Odd Polynomial Coefficients
Central Coefficient
Easy Zeroes First
Factors of x^n+1
Simple Partial Fractions
Partial Fractions with a Repeated Linear Factor
Partial Fractions with an Irreducible Quadratic

Medium (13 problems)

This tier contains 13 problems aimed at exam-standard reasoning. Representative work includes "Parity and Derivatives" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Parity and Derivatives
Behaviour Near Repeated Zeroes
Same Axes, Different Quadratics
An Odd Cubic from Two Conditions
Quadratic Roots and Symmetric Expressions
A Cubic Root Product Sum
The Opposite Roots
The Shifting Exponents
The Parity of Powers
The Cascading Powers
Sketching a Cubic with Derivatives
Divisibility by (x+1)^2
A Functional Equation for a Polynomial

Part 2 (10 problems)

Advanced (4 problems)

This tier contains 4 problems aimed at extension and synthesis. Representative work includes "The Alternating Roots of e^x" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

The Alternating Roots of e^x
The Geometric Solution of the Quartic
A Cubic with a Double Root
No Real Roots via Global Minimum

Basic (4 problems)

This tier contains 4 problems aimed at foundational fluency. Representative work includes "Multiplicity Table and Sketch" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

Multiplicity Table and Sketch
Reading a Cubic from Its Sketch
A Quartic with Paired Root Sums
Reading the Degree from a Graph

Medium (2 problems)

This tier contains 2 problems aimed at exam-standard reasoning. Representative work includes "A Family Determined by Four Intersections" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 marking expectations.

A Family Determined by Four Intersections
A Surd Expression from Quadratic Roots

Common patterns across the booklet

Vectors & geometry: 1 problem — e.g. "Legendre Polynomials and the Infinite Dot Product"
Polynomials: 19 problems — e.g. "A Factor-Theorem Identity in Three Variables"
Trigonometry: 1 problem — e.g. "The Tangent Cubic"
Sequences & series: 2 problems — e.g. "The Chebyshev Recurrence"
Other synthesis: 22 problems — e.g. "A Similar Sum-of-Squares Quartic"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

A Similar Sum-of-Squares Quartic: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Intersections of (x+1)^n and (x+1)^m: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A Factor-Theorem Identity in Three Variables: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A Common Quadratic Factor: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Intersections via P(x)-Q(x): A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
The Reversal Transformation: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

Read core formulas first; attempt before solutions
Classify: factoring, root extraction, or proof
Look for symmetry and repeated factors
Use remarks and takeaways after each solution

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Polys Ext 1 and work steadily — 45 problems is a marathon, not a sprint.

Key appendices

This booklet has no separate appendix files — formulas and takeaways are embedded in solutions and the conclusion.

Use the conclusion section as a pre-exam checklist and keep your class formula sheet nearby while practising.

Key conclusion

Polynomials reward systematic thinking — identify whether factor theorem, Vieta's relationships, or multiplicity-from-graph strategies fit the question.

The booklet's closing section reinforces these habits:

Read core formulas first; attempt before solutions
Classify: factoring, root extraction, or proof
Look for symmetry and repeated factors
Use remarks and takeaways after each solution

Revisit HSC Polys Ext 1 in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

Complete before Extension 2 Polynomials booklet; two weeks Part 1, one week Part 2 timed

General principles:

Closed-book first: attempt without notes, then check fundamentals.
Error log: record concept vs algebra vs reading errors.
Spaced repetition: redo missed questions after 3 and 7 days.
Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Polys Ext 1 booklet for?

Extension 1 students wanting focused exam-oriented polynomial practice studying Mathematics Extension 1 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Polys Ext 1 on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 45 practice problems across 70 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Extension 1 outcomes for polynomial language, remainder and factor theorems, division algorithms, sums and products of zeroes, and graph sketching. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

Applying factor theorem without checking $P (c) = 0$ at the candidate root
Confusing multiplicity with degree
Synthetic division sign errors when coefficients include zeros
Sketching end behaviour with wrong leading-term sign
Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Polys Ext 1 on Vu's Maths Hub — 45 problems with full worked solutions.

Related resources:

How to use Vu's Maths Hub — Ext 1 then Ext 2 polynomial path
HSC Polynomials — Extension 2 depth

Polynomial language and notation — exam context

In NSW Mathematics Extension 1 examinations, polynomial language and notation routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress polynomial language and notation; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Graphs of polynomial functions — exam context

In NSW Mathematics Extension 1 examinations, graphs of polynomial functions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress graphs of polynomial functions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Division algorithms — exam context

In NSW Mathematics Extension 1 examinations, division algorithms routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress division algorithms; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Remainder and factor theorems — exam context

In NSW Mathematics Extension 1 examinations, remainder and factor theorems routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress remainder and factor theorems; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Sums and products of zeroes — exam context

In NSW Mathematics Extension 1 examinations, sums and products of zeroes routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress sums and products of zeroes; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Multiple zeroes — exam context

In NSW Mathematics Extension 1 examinations, multiple zeroes routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress multiple zeroes; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Long and synthetic division — exam context

In NSW Mathematics Extension 1 examinations, long and synthetic division routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress long and synthetic division; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Graph sketching — exam context

In NSW Mathematics Extension 1 examinations, graph sketching routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Polys Ext 1 booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Polys Ext 1 and locate items that stress graph sketching; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Syllabus alignment

This booklet supports Mathematics Extension 1 under the NESA syllabus. It supplements school instruction with 70 pages of extra exam-style practice — not a replacement for class teaching.

Additional exam advice

When sitting Mathematics Extension 1 exams, allocate time proportional to marks. Practise concise justification in HSC Polys Ext 1 — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.

Why Vu's Maths Hub

Vu's Maths Hub hosts every HSC booklet in a continuous, mobile-friendly viewer — zoom, search, no download required. Maintained by Vu Hung Nguyen; CC BY 4.0 for personal and school use.

More on Remainder and factor theorems

Return to HSC Polys Ext 1 and filter mentally for remainder and factor theorems. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.

More on Sums and products of zeroes

Return to HSC Polys Ext 1 and filter mentally for sums and products of zeroes. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.

More on Multiple zeroes

Return to HSC Polys Ext 1 and filter mentally for multiple zeroes. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.

Wednesday, 1 July 2026

HSC Extension 1 Polynomials: Factor Theorem, Division, and Graphs