In this chapter you will learn how to work with algebraic expressions. You will recap some of the work on factorisation and multiplying out expressions that you learnt in earlier grades. This work will then be extended upon for Grade 10.
Recap of earlier work
The following should be familiar. Examples are given as reminders.
Parts of an expression
Mathematical expressions are just like sentences and their parts have special names. You should be familiar with the following names used to describe the parts of a mathematical expression.
Name
Examples (separated by commas)
term
,
,
,
,
,
expression
,
coefficient
,
,
,
exponent (or index)
,
base
,
,
constant
,
,
,
,
,
variable
,
equation
inequality
binomial
expression with two terms
trinomial
expression with three terms
Product of two binomials
A
binomial is a mathematical expression with two terms, e.g.
and
. If these two binomials are multiplied, the following is the result:
Find the product of
The product of two identical binomials is known as the
square of the binomial and is written as:
If the two terms are
and
then their product is:
This is known as the
difference of two squares .
Factorisation
Factorisation is the opposite of expanding brackets. For example expanding brackets would require
to be written as
. Factorisation would be to start with
and to end up with
. In previous grades, you factorised based on common factors and on difference of squares.
Common factors
Factorising based on common factors relies on there being common factors between your terms. For example,
can be factorised as follows:
Investigation : common factors
Find the highest common factors of the
following pairs of terms:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Difference of two squares
We have seen that:
Since
[link] is an equation, both sides are always equal. This means that an expression of the form:
can be factorised to
Therefore,
For example,
can be written as
which is a difference of two squares. Therefore, the factors of
are
and
.
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