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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.
The following should be familiar. Examples are given as reminders.
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 
$a\xb7{x}^{k}$ ,
$b\xb7x$ ,
${c}^{m}$ ,
$d\xb7{y}^{p}$ ,
$e\xb7y$ ,
$f$

expression  $a\xb7{x}^{k}+b\xb7x+{c}^{m}$ , $d\xb7{y}^{p}+e\xb7y+f$ 
coefficient  $a$ , $b$ , $d$ , $e$ 
exponent (or index)  $k$ , $p$ 
base  $x$ , $y$ , $c$ 
constant  $a$ , $b$ , $c$ , $d$ , $e$ , $f$ 
variable  $x$ , $y$ 
equation  $a\xb7{x}^{k}+b\xb7x+{c}^{m}=0$ 
inequality  $d\xb7{y}^{p}+e\xb7y+f\le 0$ 
binomial  expression with two terms 
trinomial  expression with three terms 
A binomial is a mathematical expression with two terms, e.g. $(ax+b)$ and $(cx+d)$ . If these two binomials are multiplied, the following is the result:
Find the product of $(3x2)(5x+8)$
The product of two identical binomials is known as the square of the binomial and is written as:
If the two terms are
$ax+b$
This is known as the difference of two squares .
Factorisation is the opposite of expanding brackets. For example expanding brackets would require
$2(x+1)$ to be written as
$2x+2$ . Factorisation would be to start with
$2x+2$
Factorising based on common factors relies on there being common factors between your terms. For example,
$2x6{x}^{2}$
Find the highest common factors of the following pairs of terms:
(a) $6y;18x$  (b) $12mn;8n$  (c) $3st;4su$  (d) $18kl;9kp$  (e) $abc;ac$ 
(f)
$2xy;4xyz$

(g) $3uv;6u$  (h)
$9xy;15xz$

(i)
$24xyz;16yz$

(j) $3m;45n$ 
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,
${x}^{2}16$
Factorise completely: ${b}^{2}{y}^{5}3ab{y}^{3}$
Factorise completely: $3a(a4)7(a4)$
Factorise $5(a2)b(2a)$
(a) $2y(y+4)$  (b) $(y+5)(y+2)$  (c) $(y+2)(2y+1)$ 
(d) $(y+8)(y+4)$  (e) $(2y+9)(3y+1)$  (f) $(3y2)(y+6)$ 
(a) $7a+4$  (b) $20a10$  (c) $18ab3bc$ 
(d) $12kj+18kq$  (e) $16{k}^{2}4k$  (f) $3{a}^{2}+6a18$ 
(g) $6a24$  (h) $2ab8a$  (i) $24kj16{k}^{2}j$ 
(j) ${a}^{2}b{b}^{2}a$  (k) $12{k}^{2}j+24{k}^{2}{j}^{2}$  (l) $72{b}^{2}q18{b}^{3}{q}^{2}$ 
(m) $4(y3)+k(3y)$  (n) $a(a1)5(a1)$  (o) $bm(b+4)6m(b+4)$ 
(p) ${a}^{2}(a+7)+a(a+7)$  (q) $3b(b4)7(4b)$  (r) ${a}^{2}{b}^{2}{c}^{2}1$ 
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