Algebraic Expressions, Binomial Expansion and Factorisation — AC9M9A02

Simplifying, expanding and factorising are connected ways of rewriting equivalent algebraic expressions. Year 9 extends these skills to binomial products and monic quadratics.

By the end of this lesson, you should be able to:

• simplify expressions by collecting like terms
• expand products using the distributive law
• expand binomial products accurately
• factorise monic quadratics and verify by re-expansion

Recall like terms, the distributive law, common factors, integer multiplication and simple linear expansion.

Simplifying preserves value

Only like terms combine. 3x²+2x−x² becomes 2x²+2x.

Expansion removes brackets by distribution

Every term in one factor multiplies every term in the other; (x+3)(x+5)=x²+8x+15.

Factorisation reverses expansion

For x²+8x+15, find integers with product 15 and sum 8: 3 and 5, giving (x+3)(x+5).

Verification is built in

Expand a factorised answer. If it does not reproduce the original expression, it is wrong.

• (x+4)²=x²+16: The middle term is 8x.
• Factor pairs only need the correct product: They must also add to the middle coefficient.
• Unlike terms can be collected: x² and x are different term types.
• Factorisation changes the value: Correct factorisation is equivalent and can be checked by expansion.

1. Simplify 4x²+3x−x²−5x.
2. Expand (x+2)(x+7).
3. Expand (x−5)(x+3).
4. Factorise x²+9x+20.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Simplify 7a−3+2a+8.
2. Expand (x−6)².
3. Expand (2x+3)(x−4).
4. Factorise x²+2x−15.
5. Factorise x²−10x+24.
6. Explain how to verify a factorised quadratic without solving an equation.

Two students factorise x²+5x−24 as (x+8)(x−3) and (x−8)(x+3). Use sum-and-product conditions to identify the correct answer without fully expanding both.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

What makes terms like?

They have the same variables raised to the same powers.

What do I look for in x²+bx+c?

Two numbers that multiply to c and add to b.

How do I check factorisation?

Expand the factors and compare with the original.

1. [4 marks] Simplify and expand a multi-part expression containing binomial products.
2. [5 marks] Factorise three monic quadratics including negative constants, showing factor-pair reasoning.
3. [6 marks] Use the side lengths x+4 and x−2 to form, expand and interpret an area expression, then solve a related condition.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

• I can collect only like terms.
• I can distribute every term across brackets.
• I can expand binomial products with correct signs.
• I can factorise monic quadratics using sum and product.
• I can verify factorisation by expansion.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

Australian Curriculum v9.0 — AC9M9A02: simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions

Victoria: VC2M9A02 — Level 9 Algebra. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Algebra and equations; Path — Further algebra and equations. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9A01 — Exponent Laws with Integer Exponents
• AC9M9A04 — Quadratic Functions, Graphs and Equations

• Australian Curriculum v9.0 — Mathematics
• Victorian Curriculum F–10 Version 2.0 — Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Simplify algebraic expressions, expand binomial products and factorise monic quadratic...

Mapped skill: simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.