AC9M10A01 · YEAR 10 · ALGEBRA

Expanding and Factorising: Two Directions of One Idea

ACARA v9 CONTENT DESCRIPTION “expand, factorise and simplify expressions and solve equations algebraically, applying exponent laws involving products, quotients and powers of variables, and the distributive property”
Builds on: Index Laws with Integer Exponents (AC9M9A01). The exponent laws from Year 9 are the engine for simplifying here. This unit puts them to work alongside the distributive property to expand, factorise, simplify, and ultimately solve equations, the core algebra that the rest of Year 10 leans on.

Two directions of one idea

A great deal of algebra rests on a single pair of opposite moves: expanding and factorising. Expanding means removing brackets by multiplying out, turning a product such as 3(x + 4) into the sum 3x + 12. Factorising goes the other way, taking a sum and writing it as a product by pulling out what its terms share. They are exact reverses, like opening and closing the same door, and that relationship is useful in itself: you can always check a factorisation by expanding it again to see if you get back where you started. Knowing which direction a problem needs, and being fluent in both, is what this unit builds toward.

Distributive law: one factor meets every term
To expand brackets, the factor outside multiplies each term inside, separately.
The distributive law says an outside factor reaches every term inside the brackets, like one person shaking hands with each. Expand to see the result.

The distributive property does the expanding

The rule that powers expanding is the distributive property: a factor outside a bracket multiplies every term inside, separately. 3(x + 4) is 3 × x + 3 × 4, which tidies to 3x + 12. The same idea handles two brackets multiplied together. To expand (x + 2)(x + 3), every term in the first bracket multiplies every term in the second, giving x², then 3x and 2x which combine to 5x, and finally 6, for x² + 5x + 6. A neat way to picture this is as the area of a rectangle whose sides are the two brackets, cut into four smaller rectangles whose areas are exactly those four products. The picture makes plain why no term may be missed.

Expanding as an area
A product of two brackets is the area of a rectangle, split into a piece for each pair of terms.
The product (x + 2)(x + 3) is the area of a rectangle split into four pieces. Reveal each piece to read off the expansion.

Factorising: finding what terms share

Factorising reverses the process by spotting a common factor. In 6x + 12, both terms are divisible by 6, since 6x is 6 × x and 12 is 6 × 2, so the expression factorises to 6(x + 2). The skill is to take out the largest factor the terms share, called the highest common factor; taking out only part of it, as in 2(3x + 6), leaves more factorising still to do. For expressions with variables, the common factor can include letters too, so that, for example, x² + x shares a factor of x and becomes x(x + 1). Whenever you factorise, expanding your answer is the quickest check that it is correct and complete.

Factorising is expanding, reversed
To factorise, find the largest factor common to every term and write it outside a bracket.
Factorising is expanding in reverse. Both 6x and 12 share a factor of 6. Pull it out to factorise.

Exponent laws keep expressions simple

Running through all of this is the need to simplify, and that is where the exponent laws from earlier do their work, now on variables as much as numbers. Multiplying powers of the same base adds the exponents, so x3 × x4 = x7. Dividing subtracts them, so x8÷ x3 = x5. Raising a power to a power multiplies them, so (x2)4 = x8. And an exponent on a bracket reaches every factor inside, so (2x3)2 = 4x6, the 2 being squared as well as the x3. These laws let a sprawling expression be written in its tidiest form, which is almost always the form you need before factorising or solving.

Exponent laws simplify expressions
Products add exponents, quotients subtract them, and a power of a power multiplies them, for variables as well as numbers.
Simplifying algebraic expressions uses the exponent laws: add exponents when multiplying like bases, subtract when dividing, multiply when raising a power to a power, and apply the power to every factor in a bracket.

Solving equations by factorising

All of these skills come together when solving equations. Many equations can be rearranged so that one side is zero and the other can be factorised into a product. The quadratic x² + 5x + 6 = 0, for instance, factorises to (x + 2)(x + 3) = 0. Now a simple and powerful idea finishes the job: if a product equals zero, then at least one of its factors must be zero. So either x + 2 = 0, giving x = −2, or x + 3 = 0, giving x = −3. The equation, which looked hard, has two clear solutions. This is the practical payoff of fluency with expanding and factorising: it turns equations into something you can solve by reasoning, step by step, with the exponent laws and the distributive property as your tools.

Solving equations by factorising
Factorise to a product equal to zero, then set each factor to zero to find the solutions.
To solve x² + 5x + 6 = 0, first factorise the left side. Reveal the solving step to finish.
Quick self-check
1. Expand 4(x + 3).
2. Expand (x + 2)(x + 3).
3. Factorise 6x + 12 fully.
4. Simplify x⁵ × x³.
5. Solve x² + 5x + 6 = 0.
Teaching pack: a week of lessons, free to printFive ready-to-teach lessons for this unit, on expanding brackets with algebra tiles, factorising, the exponent laws and solving quadratic equations by factorising: teacher plans, student worksheets with stretch sections, algebra tiles and match cards, a mini-check and every answer, with working. Print it or save as PDF, and the week is prepared.