Name of Lecturer: Mr. J.Agius
Course: HVAC2
Lesson 9
Chapter 3: Algebra

The Product of two Binomial Expressions
A Binomial Expression consists of two terms. Thus 3x + 5, a + b, 2x + 3y and 4p – q are all
binomial expressions.
To find the product of (a + b)(c + d) consider the following diagram.
a+b
b
a
P
A
c+d
c
Area = ac
Area = bc
T
R
Area = ad
d
B
S
Area = bd
Q
D
C
In this diagram the rectangular area ABCD is made up as follows:
ABCD = APTR + PBST + RTQD + TSCQ
i.e.
(a + b)(c + d) = ac + ad + bc + bd
It will be noticed that the expression on the right hand side is obtained by multiplying each
term in the one bracket by each term in the other bracket. The process is illustrated below:
(a + b)(c + d) = ac + ad + bc + bd
Example 1
Expand:
a
(3x + 2)(4x + 5)
b
(2p – 3)(4p + 7)
c
(z – 5)(3z – 5)
d
(2x + 3y)(3x – 2y)
Answer
a) (3x + 2)(4x + 5)
3 Algebra
= 3x × 4x + 3x × 5 + 2 × 4x + 2 × 5
= 12x2 + 15x + 8x + 10
= 12x2 + 23x + 10
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Name of Lecturer: Mr. J.Agius
Course: HVAC2
b) (2p – 3)(4p + 7)
= 2p × 4p + 2p × 7 – 3 × 4p – 3 × 7
= 8p2 + 14p – 12p – 21
= 8p2 + 2p – 21
c) (z – 5)(3z – 5)
= z × 3z + z × (– 5) + (– 5) × 3z + (– 5) × (– 5)
= 3z2 – 5z – 15z + 25
= 3z2 – 20z + 25
d) (2x + 3y)(3x – 2y) = 2x × 3x + 2x × (– 2y) + 3y × 3x + 3y × (– 2y)
= 6x2 – 4xy + 9xy – 6y2
= 6x2 + 5xy – 6y2

The Square of a Binomial Expression
(a + b)2 = (a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2
The square of a binomial expression is the sum of the squares of the two terms and twice
their products.
(a – b)2 = (a – b)(a – b) = a2 – ab – ba + b2 = a2 – 2ab + b2
Example 2
Find the products of the following:
(2x + 5)2
a
(3x – 2)2
b
c
(2x + 3y)2
Answer

a)
(2x + 5)2
= (2x + 5) (2x + 5)
= 4x2 + 20x + 25
b)
(3x – 2)2
= (3x – 2) (3x – 2)
= 9x2 – 12x + 4
c)
(2x + 3y)2
= (2x + 3y) (2x + 3y) = 4x2 + 12xy + 9y2
The Product of the Sum and Difference of Two Squares
(a + b)(a – b) = a2 – ab + ba – b2
= a2 – b2
This result is the difference of the squares of the two terms.
Example 3
Expand:
a
(8x + 3)(8x – 3)
b
(2x + 5y)(2x – 5y)
Answer
a) (8x + 3)(8x – 3)
= 64x2 – 9
b) (2x + 5y)(2x – 5y) = 4x2 – 25y2
3 Algebra
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Name of Lecturer: Mr. J.Agius
Course: HVAC2
Products of Binomial Expressions
Q1
Find the products of the following:
a
(x + 1)(x + 2)
b
(x + 3)(x + 1)
c (x + 4)(x + 5)
d
(2x + 5)(x + 3)
e
(3x + 7)(x + 6)
f
(5x + 1)(x + 4)
g
(2x + 4)(3x + 2)
h
(5x + 1)(2x + 3)
i
(7x + 2)(3x + 5)
j
(x – 1)(x – 3)
k
(x – 4)(x – 2)
l
(x – 6)(x – 3)
n
(x – 2)(3x – 5)
o (x – 8)(4x – 1)
m (2x – 1)(x – 4)
p
(2x – 4)(3x – 2)
q
(3x – 1)(2x – 5)
r
s
(x + 3)(x – 1)
t
(x – 2)(x + 7)
u (x – 5)(x + 3)
v
(2x + 5)(x – 2)
w (3x – 5)(x + 6)
Q2
(7x – 5)(3x – 2)
x (3x + 5)(x + 6)
Expand:
a
(3x + 5)(2x – 3)
b
(6x – 7)(2x + 3)
c (3x – 5)(2x + 3)
d
(3x + 2y)(x + y)
e
(2p – q)(p – 3q)
f
(3v + 2u)(2v – 3u)
g
(2a + b)(3a – b)
h
(5a – 7)(a – 6)
i
(3x + 4y)(2x – 3y)
j
(x + 1)2
k
(2x + 3)2
l
(3x + 7)2
m (x – 1)2
n
(3x – 5)2
o (2x – 3)2
p
(2a + 3b)2
q
(x + y)2
r
s
(a – b)2
t
(3x – 4y)2
u (2x + y)(2x – y)
v
(a – 3b)(a + 3b)
w (2m – 3n)(2m + 3n)
Q3
(p + 3q)2
x (x2 – y)(x2 + y)
Multiply out the brackets then simplify your expressions where possible.
a (a + 1)(b + 2) – ab
b (2p + 1)(q – 2) + 3p – pq
c (3r – s)(r – 1) + s – 2r
d (2p + 3a)(2p – 3a) + 9a2
3 Algebra
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