ALGEBRAIC EXPRESSIONS
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Real numbers
Surds lie between integers
Rounding-off Integers
Products
Factorization: Common Factor
Factorization: Difference of Squares
Factorization: Trinomials
Factorization: Grouping in Pairs
Factorization: Cubes
Algebraic Fractions
Algebraic Language
1
REAL NUMBERS
Rational Numbers
A rational number is a number that can be
expressed in the form
a
b
where b  0
and where a and b are integers.
2
Examples
◼
a) Integers
5
e.g. 5 can be written as
where 5 and 1 are
1
integers.
◼
b) Mixed fractions
e.g. 2 3
7
◼
c) Terminating decimals
e.g. 0,25 = 25 = 1
100
◼
4
d) Recurring decimals have an infinite pattern & can
be expressed as a fraction
e.g. 0, 3 = 0, 3333333; 0, 12 = 0, 12121212
3
Converting Recurring Decimals to Fractions
◼
▪
E.g. a) Show that 0,3 is rational.
Let x = 0, 333333....
l0x = 3, 333333 (multiply both sides by 10)
l0x - x = 3, 33333 – 0, 33333 (subtract equations)
9x = 3, 0000000
9x = 3
x = 3 … a rational number!
E.g. b) Show that 0, 12 is rational.
Let x = 0, 12121212
100x = 12, 12121212 (multiply both sides by 100)
99x = 12, 000000 (subtract equations)
12 4
x = 99 = 33 … a rational number!
4
EXERCISE
1. Are these numbers rational and why?
(a)
(b)
1
3
4
−2
1
6
(c) 6
(d) - 3
(e) 0, 72
(f) 1, 4142
5
2. Show that the following recurring decimals
are rational:
(a) 0, 4
(b) 0, 21
(c) 0, 14
(d) 19, 45
(e) 0, 124
(f) 0, 124
6
Irrational numbers
◼
◼
◼
Numbers that cannot be written in the form
where b  0
Therefore recurring numbers that neither
terminate nor recur with a pattern
E.g.
a) 5,739129…
b) -4,883291103…
c) 
a
b
Irrational Numbers in Circles & Squares
7
The Number
◼
◼


is the ratio of the circumference of a circle to
its diameter
 is = 3,142857143….
Rounding-off π
◼
However,  can be approximated as an
improper fraction 22
7
π as a Rational Number
8
EXERCISE 1
State whether the following numbers are
rational or irrational:
(a) 8
(b) 16
(c) 7
(d) 7
1
(e) 25
(f) 0
(g) 5 2
(h) − 1 16
(i) - 0, 13
(j) 0, 42 (k) 
(l) 0, 2453756…
9
EXERCISE 2
Classify numbers by placing ticks in the appropriate
columns:
Number
−3
Real
Rational
Integer
Whole
Natural
Irrational
1
6
0,14674
16
5
-3

22
7
0, 3
8, 23647
10
SURDS LIE BETWEEN INTEGERS
E.g. Determine without the use of a calculator,
between which 2 integers 11 lies.
◼ Find an integer smaller and bigger than 11
that can be square rooted … 9 and 16
◼ Now create an inequality … 9 < 11 < 16
◼ Square root all integers …
9  11  16
◼ Solve …
 3  11  4
11 = 3,3166
◼ Check using a calculator …
11
EXERCISE
Without using a calculator, determine between
which two integers the following irrational
numbers lie:
(a)
(b)
(c)
(d)
(e)
(f)
30
27
3
43
− 6
7
33

12
ROUNDING-OFF INTEGERS
◼
◼
If it is > 5 or = 5 … round up
If it is < 5
… round down
◼
Remember! If you are rounding-off to 2 decimal
places, the third decimal place determines whether
you round up or down etc.
◼
E.g. (a) 2, 31437 (2 d.p.) …
Answer:2, 31
(b) 0, 77777 (3 d.p.) …
Answer: 0, 778
(c) 245, 13589 (4 d.p.) … Answer: 245,1359
Rounding-off Numbers
13
EXERCISE 1
Round off the following numbers to the
number of decimal places indicated:
◼ (a) 9, 23584
(3 decimal places)
◼ (b) 67, 2436
(2 decimal places)
◼ (c) 4, 3768534
(4 decimal places)
◼ (d) 17,247398
(5 decimal places)
◼ (e) 79, 9999
(3 decimal places)
◼ (f) 34, 2784682
(4 decimal places)
◼ (g) 5,555555
(5 decimal places)
14
EXERCISE 2
Simplify and round-off to the number of
decimal places indicated:
▪
(a) 7,53427
(3 decimal places)
▪
(b) 3,3333
(4 decimal places)
▪
(c) 36,268
(2 decimal places)
▪
(d) (2,64) (2,18)
(5 decimal places)
▪
(e) (1,64)  1,64
(2 decimal places)
3
6
2
6
6
15
PRODUCTS
◼
◼
E.g. x (y + z) = xy + xz
- Multiply each term inside the bracket by the
number outside the bracket
E.g. (a + b)(c + d) = ac + ad + bc + bd
- This is done by using the FOIL method
The Product Game
16
Examples
Expand and simplify the following:
◼ (a) (x + 2) (x + 3)
= x + 3x + 2 x + 6
2
= x + 5x + 6
2
◼
Squaring a Binomial Example
(b) (x + 2) (x² + x - 1)
= x + x − x + 2x + 2x − 2
3
2
2
= x 3 + 3x 2 + x − 2
17
▪ (c) x(x²-2xy+3y²) - 2y(x² -2xy+3y²)
= x( x 2 − 2 xy + 3 y 2 ) − 2 y ( x 2 − 2 xy + 3 y 2 )
= x 3 − 2 x 2 y + 3xy 2 − 2 x 2 y + 4 xy 2 − 6 y 3
= x 3 − 4 x 2 y + 7 xy 2 − 6 y 2
▪ (d) (a – 3b) (a – 3b)²
= (a − 3b)(a − 3b)
2
= (a − 3b)(a 2 − 6ab + 9b 2 )
= a(a − 6ab + 9b ) − 3b(a − 6ab + 9b )
2
2
2
2
= a 3 − 6a 2 b + 9ab 2 − 3a 2 b + 18ab 2 − 27b 3
= a − 9a b + 27ab − 27b
3
2
2
3
18
Exercise 1:
4
Simplify:
◼ (a) (x + 3)(x - 3)
◼ (b) (x - 6)(x + 6)
◼ (c) (2x - l)(2x + l)
◼ (d) (4x + 9)(4x - 9)
◼ (e) (3x - 2y)(3x + 2y)
◼ (f) (4a³ b + 3)(4a³ b - 3)
◼ (g) (2x – 3 + y)(2x – 3 – y)
◼ (h) (1 – a )(1 – a )(1 + a)
19
Exercise 2
2
Simplify:
◼ (a) 2x(3x - 4y)² - (7x - 2xy)
◼ (b) (5y + 1)² - (3y + 4)(2 - 3y)
◼ (c) (2x + y) - (3x - 2y) + (x - 4y)(x + 4y)
◼ (d) (8m - 3n)(4m + n) - (n - 3m)(n + 3m)
◼ (e) (3a + b)(3a - b)(2a + 5b)
20
Exercise 3
Simplify:
◼ (a) (x + 1)(x² + 2x + 3)
◼ (b) (x - 1)(x² - 2x + 3)
◼ (c) (2x + 4)(x² - 3x + 1)
◼ (d) (2x - 4)(x² - 3x + 1)
◼ (e) (3x-y)(2x² + 4xy – y² )
◼ (f) (3x - 2y)(9 x² + 6xy + 4 y² )
◼ (g) (3x + 2y)(9x² - 6xy + 4y² )
◼ (h) (2a + 3b)²
◼ (i) (2a² - 3b)²
21
FACTORIZATION: Common Factor
The Factor Game
◼
The golden rule of factorization is to always look for the
highest common factor first:
Basic examples
Common Factor
with Variables
Complex example
e.g. a(x-y) – 2(x-y)² = (x-y)[a-2(x-y)]
= (x-y)(a-2x+2y)
22
Common Brackets Exercise:
◼
◼
◼
◼
◼
◼
◼
◼
(a) a(x + y ) + b(x + y )
(c) p(q + r ) + m(r + q )
(e) (x − y ) − 3(x − y )
(g) (m − n) + (m − n)
(i) 7 x(m − 3n) + 4 y(3n − m)
(k) 2x(3 p + q) + 4 y(− q − 3 p)
(m) 4x(a − 2) + (2 − a )
(o) (a − 3b) − c(3b − a) + d (3b − a)
2
6
3
2
3
(b) x(a + b) + y(a + b)
(d) 2 x(m − 3n) − 5 y(m − 3n)
(f) (a + c ) + (a + c )
(h) 7 x(m − 3n) − 4 y(3n − m)
(j) 7 x(m + 3n) + 4 y(− m − 3n)
(l) (a − b ) − ( pb − pa )
(n) 2x (3a − b) − 12x(b − 3a)
4
5
2
23
FACTORIZATION: Difference of
Squares
There must be 2 terms that you can take the square-root of and
a minus sign.
Basic examples
▪ a) a − b =
2
2
( a − b )( a + b )
2
2
2
2
= (a − b)(a + b)
▪ b) x 2 − 9
= ( x 2 − 9 )( x 2 + 9 )
= ( x − 3)( x + 3)
Difference of Squares Example
24
Complex examples
▪ a)
49x 4 − 64 y 2
(
)(
= 7x 2 − 8y 7x 2 + 8y
▪ b)
)
4 − (x − y )
2
= [2 + (x − y )][2 − (x − y )]
= (2 + x − y )(2 − x + y )
▪ c)
8a 8 − 8b 8
(
= 8(a
= 8(a
= 8 a 8 − b8
4
− b4
2
− b2
)
)(a
)(a
4
+ b4
2
+ b2
(
)
)(a
4
+ b4
)(
)
= 8(a − b )(a + b ) a 2 + b 2 a 4 + b 4
)
25
Complex Exercise
◼
◼
◼
◼
◼
◼
◼
◼
(a) 100m 8 − n 6
(b) 144x − 225y
(c) 12k − 75m
(d) p − 16
(e) z − 81
(f) 27a − 3ab
(g) (x + b) − c
(n) 25a − 16(a − m)
4
4
2
2
4
8
3
2
2
2
2
2
26
FACTORIZATION: Trinomials
Make sure you know your times-tables and factors!
◼
◼
◼
◼
◼
◼
◼
◼
◼
E.g. a) a 2 + 6a + 8
Factors of 8: 1 x 8 or 4 x 2
The middle term (6a) is obtained by adding the
factors of 8 … 4 + 2 = 6
Therefore: = (a + 4)(a + 2)
E.g. b) 3x 2 − 21x − 24
2
=
3
(
x
− 7 x − 8)
First take out common factor!
Factors of 8: 1 x 8 or 4 x 2
The middle term (7x) is obtained by adding the
factors of 8 … -8 +1 = -7
Therefore: = 3( x 2 − 7 x − 8)
Trinomial with Common Factor
= 3( x − 8)( x + 1)
27
Note:
◼
If the sign of the last term of a trinomial is
positive, the signs in the brackets are the
same
i.e. (… - …)(… - …) or (… + …)(… + …)
◼
If the sign of the last term of a trinomial is
negative, the signs in the brackets are
different, i.e. both positive or both negative
i.e. (… + …)(… - …) or (… - …)(… + …)
Visualizing Factorization
28
Basic Exercise
Factorize fully :
◼
◼
◼
◼
◼
◼
◼
a 2 + 7a + 12
(a)
(c) a 2 + 4a − 12
(e) x 2 − 9 x + 20
(g) x 2 − 12 x + 35
(i) p 2 + 5 p − 6
(k) k 2 − 11k + 28
(m) k 2 + 6k + 9
a 2 − 7a + 12
(b)
2
a
(d) − 4a − 12
2
x
− 11x − 12
(f)
2
p
(h) − 5 p − 6
(j) p 2 − 5 p + 6
(l) k 2 − 5k − 84
(o) k 2 − 8k + 16
29
Complex Exercise
Factorize fully:
◼
◼
◼
◼
◼
(a)
(b)
(c)
(d)
(e)
3k 2 − 3k − 18
2k 2 − 14k − 36
4k 2 + 12k − 40
6 g 2 + 24g − 30
k 3 + 11k 2 − 8k
30
More advanced trinomials
E.g. a) 21p 2 + 25 p − 4
◼
◼
◼
Step 1: Check for the HCF … none
Step 2: Write down the brackets and the factors of
the first term and the factors of the last term …
(7p 1)(3p
4)
Step 3: Now multiply the innermost and the
outermost terms … 1  3 p = 3 p
7 p  4 = 28 p
Step 4: To find the middle term ... - 3p + 28p = + 25p
◼ Step 5: Complete the factors … (7p – 1)(3p + 4)
Note! This method involves trial and error and you need to keep t
◼
trying different options until you get ones that will work.
31
E.g. b) 24a 2 − 10ab − b 2
◼
◼
◼
◼
◼
Step 1: Check for the HCF … none
Step 2: Write down the brackets and the factors of the
first term and the factors of the last term …
(12a
b)(2a
b)
Step 3: Now multiply the innermost and the outermost
terms … 2ab x 12ab
Step 4: To find the middle term ... -12ab + 2ab = - 10ab
Step 5: Complete the factors … (12a + b)(2a – b)
32
Advanced Exercise
1. Factorize fully:
◼
◼
◼
◼
◼
◼
(a) 2 p + 3 p + 1
(c) 15 p 2 − 8 p + 1
(e) 2m 2 − 5m − 3
(g) 2m 2 − 7m + 6
(i) 6k 2 − 5k − 21
(k) 18k 2 − 3k − 10
2
3p2 − 4 p +1
(b)
(d) 18m 2 + 3m − 1
(f) 5m 2 + 14m + 8
(h) 6k 2 − 11k − 10
(j) 20k 2 + 24k − 9
(l) 15 − x − 6 x 2
33
FACTORIZATION: Grouping in pairs
Group terms with common factors or similar brackets!
ax + ay + px + py
E.g.
◼ Group the terms that look similar
(i.e. those that could potentially have
common factors)
◼ Factorize each pair separately and then take
out the common bracket:
ax + ay + px + py
= a( x + y ) + p( x + y )
= ( x + y )(a + p )
34
Switch-arounds
“taking out a negative”
◼ - x + y = - (x - y) and – x – y = - (x + y)
E.g. a) Factorize
a 2 + a − 6ax − 6 x
= (a 2 + a ) + (−6ax − 6 x)
= a (a + 1) − 6 x(a + 1)
= (a + 1)(a − 6 x)
35
◼
b) Factorize:
p3 − 3 p 2 − p + 3
= ( p 3 − 3 p 2 ) + ( − p + 3)
= p 2 ( p − 3) − ( p − 3)
= ( p − 3)( p 2 − 1)
= ( p − 3)( p − 1)( p + 1)
◼
(c) Factorize: 6m2 + 3m − 6 p −12mp
= (6m 2 + 3m) + (−6 p − 12mp)
= 3m(2m + 1) − 6 p(1 + 2m)
= 3m(2m + 1) + 6 p(2m + 1)
= (2m + 1)(3m + 6 p)
= 3(2m + 1)(m + 2 p)
36
Exercise
Factorize:
◼ (a) 6a 3 − 2a 2 − 54a + 18
◼ (b) p 2 − (d + t ) p + dt
◼ (c) m 2 − 9 − (m − 3)(1 − 2m)
◼ (d) x 2 − y 2 − x − y
◼ (e) p 2 − 4 pq + 4q 2 − 16t 2
37
FACTORIZATION: Cubes
Sum of Cubes
x.x²=x³
y.y²=y³
x³ + y³ = (x+y)(x² - xy + y²)
Take the cube root
of each term
Times factors of
first bracket to
get middle term
Sum of Cubes Example
38
Difference of Cubes
x.x²=x³
y.y²=y³
x³ - y³ = (x-y)(x² + xy + y²)
Take the cube root
of each term
Difference of Cubes Example
Times factors of
first bracket to
get middle term
Visualization of Factorizing a Cubic Expression
39
ALGEBRAIC FRACTIONS
Simplify the following expressions:
◼
(a)
◼
(b)
12a 3 b 4 t
18a 5 bt
2
7
12a c 24a c

5d
25d
2a 3−5 b 4 −1t 1−1
=
3
2a − 2 b 3 t 0
=
3
2b 3
=
3a 2
=
=
=
=
12a 2 c
25d

5d
24a 7 c
12  25a 2 cd
5  24a 7 cd
5a 2 − 7 c 1−1 d 1−1
2
5
2a 5
40
Whenever the numerator contains two or more terms,
factorize the expression in the numerator and simplify
◼
(c)
6c 2 + 12c
6c 2
◼
(d)
9x2 −1
3x − 1
6c (c + 2)
6c 2
c+2
=
c
=
(3 x + 1)(3 x − 1)
=
3x − 1
= 3x + 1
Simplifying Basic Algebraic Expressions
41
EXERCISE 1
Simplify the following:
◼
(a)
◼
(c)
◼
(e)
◼
(g)
◼
(i)
24 x 6 y 10 z
36 x 8 yz
2
5
8m

2m
25
2x3 6 y 4x3


y
3y 2 x
4p
−8p
4p
2
4t 2 + 4m
4m 2
(b)
25 x 3 y 5 z 2
75 xy 6 z 2
(d)
6t 2
4t 3

p
p2
(f)
m2 − m
m
(h)
3s 2 − 6 s
6s
(j)
w3 − w 2
w2

4
8
42
EXERCISE 2
Simplify
◼
◼
◼
2w 1
(a)
+
3 6
(c) 3 + 2
t t2
(e) 7
2
1
−
+ 3
6r 9rt 3r
c c
(b) −
3 4
(d) 5 − 3 + 1
w3 w 2
(f)
3
5
7
−
+
−4
2
2
2 y w 4 yw 6 y
43
More Advanced Algebraic Fractions
Examples
a)
b)
( x − 5)( x − 1) + 2( x − 5)
x −5
xy − x 2
x2
 2
2
2
y −x
y + xy
Simplifying Complex
Algebraic Expressions
( x − 5)[( x − 1) + 2]
=
x −5
= ( x − 1) + 2
= x +1
xy − x 2
y 2 + xy
= 2

2
y −x
x2
x( y − x)
y ( y + x)
=

( y − x )( y + x )
x2
y
=
x
44
Exercise
Simplify:
a)
x 2 + 5x + 4
x ( x + 1)
b)
x 2 ( x − 1) + x 2 − 1
x ( x − 1)
c)
x2 + x − 2
2+ x

2( x − 2)
6 − 3x
d)
1− 2x
x+4
1
−
+
4 x 2 − 1 2 x 2 − 3x + 1 1 − x
45