Algebra
I. Simplifying
Simplifying algebra involves solving as much of the problem as you can.
a) Addition
When adding a problem, or expression, involving algebra i.e. pronumerals amongst
the terms, only the like terms can be added. Like terms mean terms with the same
pronumeral with the same power or root values.
For example:
2a  b  5b 2  2b  4a
 6a  3b  5b 2
b) Subtraction
Subtraction works the same as addition, with only the like terms being able to be
subtracted
from one another.

For example:
2a  b  5b 2  2b  4a
 2a  3b  5b 2
When an expression involves both adding and subtracting, the same laws apply as
normal addition and subtraction, but the like terms rule still apply.
 c) Multiplication
When multiplying algebraic expression, like terms are not the only terms that can be
multiplied by one another. If two expressions with different pronumerals are
multiplied, then after the numerals in the solution come the pronumerals from the
question in alphabetical order, one after the other. This shows that they have been
multiplied.
7a  2v  m
For example:
 14amv
When multiplying algebraic expressions with the same pronumeral, however many
times this pronumeral is multiplied it ends up as a power.
2a  3a  2b  a  b

For example:
12a3b2
d) Division
Division with algebraic expressions is easier when done in a fraction layout, i.e. with
one 
value over another.
3a 2b 2  6ab
For example:
3a 2b 2
6ab
With this layout, it is now possible to cancel out and simplify the problem as far as
possible.


3a 2b 2
6ab
3 a  a b b

6ab
ab

2
This solution is found because 3 and 6 cancel to give 1 and 2, and the a and b on the
bottom cancelled one a and one b on top, leaving one of each.

II. Factorisation
Factorization is the resolution of number/s into factors so that when multiplied
together they give those number/s.
a) Simple
Simple factorisation involves expressions that have the common factor outside of the
brackets containing the numbers that have had the common factor taken out of them
To factorise a simple expression, the distributive law: ax  bx  x(a  b) .
Example: x 2  2x  x(x  2) because the common factor is x. And:
6a2b3  3a3b2  3a2b2 (2b  a) similarly.

b) Grouping in pairs

When
an expression has for terms, then it can be factorised into pairs. The standard

ax  bx  ay  by
form for this is  x(a  b)  y(a  b) .
 (a  b)(x  y)
2x  4  6y  3xy
 2(x  2)  3y(x  2)
Example:
  2(x  2)  3y(x  2)
 (x  2)(2  3y)
c) Mixed
Mixed factors are where an expression needs more then one method of factorising to
complete
it. They are case specific with combinations of the methods above.

24  3b 3  3(8  b 3 )
Example:
This uses simple factors and the difference of 2
 3(2  b)(4  2b  b2)
cubes.
d) Trinomials
A
trinomial is an expression with three terms. Factorising one usually gives a
binomial product. The standard form for a trinomial is
x 2  (a  b)x  ab  x 2  ax  bx  ab
 x(x  a)  b(x  a)
 (x  a)(x  b)

.
x 2  5x  24  x 2  8x  3x  24
Example:  x(x  8)  3(x  8)
because a in the standard form is –8 here,
 (x  8)(x  3)
and b in standard form is 3.
e) Perfect squares
Perfect squares are the factorising of binomial products to get grouping in

pairs, except that the pairs are the same so that they can be displayed as one
expression in brackets and squared. The standard from for perfect squares are
a2  2ab  b2  (a  b)2 and a2  2ab  b2  (a  b)2 .
Example: 4x 2 12x  9  (2x  3)2


p2  2 5p  5  (p  5)

f) Difference of 2 squares

The standard form for the difference of 2 squares is a2  b2  (a  b)(a  b) .
Example: 4a 2  3  (2a  3)(2a  3)
g) Sum of 2 cubes
 a3  b3  (a  b)(a2  ab  b2 ) . Here, the
The standard form for the sum of 2 cubes is

acronym
of S.O.P.S can be applied, which stands for square of first pronumeral,
opposite sign between pronumerals, both pronumerals together and then square of the
last pronumeral.

Example 8  27y 3  (2  3y)(3  6y  9y 2 )
h) Difference of 2 cubes
The standard form for the difference of 2 cubes is a3  b3  (a  b)(a2  ab  b2 ) .

Once again, S.O.P.S can be applied.
Example: a3b3 1  (ab 1)(a2b2  ab 1) and
2
8(a  3) 3  b 3  (2a  6  b)(4a 2  12a  12a
  36  2ab  6b  b )
because the first
 (2a  6  b)(4a 2  24a  2ab  6b  b 2  36)
cube
 root, 2a+6, is one expression and must be square out as the whole thing, like a
binomial product.

III. Expansion
Expansion is the process of removing grouping symbols, such as brackets or
parentheses, so that the expression can be seen in it whole form.
a) Simple
Simple expansion involves the use of the distributive law a(b  c)  ab  ac . This
removes the brackets and keeps the expression true.
4  3(a  5)  (a  7)  4  3a  15  a  7
Example: 2(4 y  3)  8y  6 and
 28  2a 
b) Binomials
Binomials refer to two numbers, but in reference to expansion we refer to binomial

products, with two binomials
 been multiplied together.
i) Simple
Simple binomials involve the two terms in the first bracket being multiplied by the
two terms in the second brackets. The standard form for simple binomials is
(a  b)(x  y)  ax  ay  bx  by .
Example:

(2y  3)(y  5)  2y 2  10y  3y 15
 2y 2  7y 15
ii) Perfect squares
When some binomial products are expanded, we can see that they have special
properties
that are used in other areas and that they can be expanded or factorised

quickly.
Perfect squares, in reference to expansion, are binomial products have the same
expressions multiplied by one another, i.e. one binomial squared. Their standard form
is (a  b)2  a2  2ab  b2 and (a  b)2  a2  2ab  b2 .
Example: (t  4)2  t 2  8t 16 and (3y 1) 2  9y 2  6y  1


IV. Other Algebra functions
There are other uses 
for algebra, which can be seen below.

a) Substitution 
Algebra is used in formulae so that when measurements are taken the pronumerals
can be substituted for real numbers. Some formulae included finding the area of a
1
triangle A  bh , where A = area, b= base and h= height. When measurements are
2
found for b and h, the area of the triangle can be found.
b) Algebraic Fractions
Algebraic fractions involve expressions with known and unknown denominators and
the four operations (addition, subtraction, multiplication and division).
When there is a known denominator, the fractions are simplified the same way as
ordinary fractions.
When an unknown denominator is present and multiplication or division is required,
they are also done the same way as ordinary fractions.
When an unknown denominator is present and addition or subtraction is needed, then
the same thing can be done as with normal fractions, that is, finding the lowest
common denominator. This usually finds a binomial product or other algebraic
expressions before the addition and subtraction is done. All different scenarios have
examples below.
x 1 x  3

5
4
4 x  4  5x 15 2a 2  10ab a 2  25


b 3  27
4b  12
20
x 19

20
c)Completing the square
Completing the square is a method derived from a perfect square. The general form
when factorising a perfect square is a2  2ab  b2  (a  b)2 .

When given an expression in the form of a 2  pa, we can complete the square by
p
p
halving p and squaring it. Thus it gives a 2  pa  ( ) 2 . Since p  2b, b  and
2
2
p
to give a perfect square and another number (unless
b2  ( )2 . This can be factorised
2
it is a quadratic equation.



