Expanding and Factorising: Two Directions of One Idea
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.
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.
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.
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.
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.