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 three times the bracket x plus four into the sum three x plus twelve. 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. Three times the bracket x plus four is three times x plus three times four, which tidies to three x plus twelve. The same idea handles two brackets multiplied together. To expand the product of x plus two and x plus three, every term in the first bracket multiplies every term in the second, giving x squared, then three x and two x which combine to five x, and finally six, for x squared plus five x plus six. 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 six x plus twelve, both terms are divisible by six, since six x is six times x and twelve is six times two, so the expression factorises to six times the bracket x plus two. 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 two times the bracket three x plus six, leaves more factorising still to do. For expressions with variables, the common factor can include letters too, so that, for example, x squared plus x shares a factor of x and becomes x times the bracket x plus one. 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 x cubed times x to the fourth is x to the seventh. Dividing subtracts them, so x to the eighth divided by x cubed is x to the fifth. Raising a power to a power multiplies them, so the bracket x squared all to the fourth is x to the eighth. And an exponent on a bracket reaches every factor inside, so the bracket two x cubed all squared is four x to the sixth, the two being squared as well as the x cubed. 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 x squared plus five x plus six equals zero, for instance, factorises to the bracket x plus two times the bracket x plus three equals zero. 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 plus two is zero, giving x equals minus two, or x plus three is zero, giving x equals minus three. 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 x squared plus 5x plus 6 = 0, first factorise the left side. Reveal the solving step to finish.