Year 10 Mathematics · AC9M10A01

Expanding, Factorising, Simplifying and Solving Algebra

Build the algebra fluency needed across Year 10: apply exponent laws correctly, move between expanded and factorised forms, simplify expressions without changing their…

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Key conceptTeach from the board

1. Equivalent algebraic forms

Expressions can look different and still be equal for every permitted value of the variable. For example, 3(x + 4) and 3x + 12 are equivalent. Expansion changes a product into a sum; factorisation reverses that process.

2. Exponent laws require matching bases

For non-zero a where required: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ and a⁰ = 1. A negative exponent means a reciprocal: a⁻ⁿ = 1/aⁿ. These laws apply to multiplication, division and powers—not to addition.

3. Expanding

Distribute every factor to every required term. For binomials, (x + a)(x + b) = x² + (a+b)x + ab. A reliable method matters more than memorising a visual acronym: every term in one bracket must multiply every term in the other.

4. Factorising

First look for a greatest common factor. For x² + bx + c, find numbers p and q with pq = c and p + q = b. Then x² + bx + c = (x+p)(x+q). Always expand the result to check it.

5. Solving equations

An equation states that two expressions have the same value. Legal algebraic steps preserve that equality. For a factorised quadratic, the null factor law says that if AB = 0, then A = 0 or B = 0.

Worked examplesWe do

Worked examples

AC9M10A01 - Expanding, Factorising, Simplifying and Solving Algebra
Example 1

Simplify x⁵ × x⁻². Which exponent law applies?

Example 2

Factorise x² − x − 12. Find a product of −12 and sum of −1.

Example 3

Simplify a⁴a⁻⁷ with positive exponents.

Example 4

Name one reliable check for each of Questions 2–4.

Curriculum examplesCopied content

Australian Curriculum:AC9M10A01, Year 10 Algebra — expanding, factorising and simplifying expressions and solving equations algebraically using exponent laws and the distributive property.

Victoria: Level 10 divides this broad Australian descriptor across several descriptors: VC2M10A01 common factors, VC2M10A02 exponent laws, VC2M10A04 binomial expansion/monic quadratic factorisation, VC2M10A07 linear equations, VC2M10A12 linear equations with simple algebraic fractions and VC2M10A13 simple quadratic equations. The mapping is therefore strong collectively but not one-to-one.

NSW: Stage 5 spreads this content across Core and Paths. Primary overlaps include MA5-ALG-C-01, MA5-IND-C-01, MA5-EQU-C-01 and, for extended algebra, MA5-ALG-P-01/02, MA5-IND-P-01 and MA5-EQU-P-01/02. Schools may sequence Core and Path content differently across Years 9 and 10.

Alignment explanation: The exponent-law, expansion, factorisation and equation sections collectively match AC9M10A01. Victoria and NSW unpack those skills into narrower outcomes, so SkillrHub maps each lesson component to the relevant state content instead of pretending a single state outcome is equivalent.

Lesson componentAustralian CurriculumVictoriaNSW
Exponent lawsAC9M10A01VC2M10A02MA5-IND-C-01; MA5-IND-P-01 supporting Path
Expand and simplifyAC9M10A01VC2M10A04MA5-ALG-C-01; MA5-ALG-P-01/02
FactoriseAC9M10A01VC2M10A01, VC2M10A04MA5-ALG-P-01/02
Solve equationsAC9M10A01VC2M10A07, A12, A13MA5-EQU-C-01; MA5-EQU-P-01/02
Reason and verifyAC9M10A01 + proficiencyLevel 10 mathematical proficiencyMAO-WM-01
Questions and answersWith answers
Why factorise?
Factorisation can expose common structure, simplify later work and turn a quadratic equation into factors that can be solved using the null factor law.
How do I check a factorisation?
Expand it. If you recover the original expression exactly, the factorisation is correct.
When can exponents be added?
When multiplying powers with the same base.
Why do negative exponents create reciprocals?
The quotient law must remain consistent: a²/a⁵ = a⁻³ and also equals 1/a³.
How do I check an equation solution?
Substitute it into the original equation and confirm both sides are equal.
Practice and reviewReady for practice
  • x² + x³ = x⁵. False. Exponents add when multiplying the same base: x² × x³ = x⁵.
  • (a+b)² = a²+b². False. Expanding gives a² + 2ab + b².
  • Only the first term gets multiplied. In 4(2x−3), both 2x and −3 are multiplied by 4.
  • A factor pair only needs the right product. For a monic quadratic it must also have the correct sum.
  • Dividing by x is always safe. Dividing by a variable can lose a solution when that variable could be zero. Factor first when appropriate.
  • Solving ends when an expression looks simpler. A solution must be a value that satisfies the original equation.
Curriculum alignmentStart here
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