1. Revision - Algebra I Binomial Product
Using Distributive Law to
Expand / Remove Brackets.
By I Porter

2. Definitions Binomial Products
Grouping symbols (…..) can be expanded by using the Disributlive Law.
This states that for any real numbers a, b and c:
a(b ± c) = ab ± ac
Examples:
5(2+3) = 5 x x 3 3(7 - 4) = 3 x x 4
5 x 5 = x 3 =
25 = 25 9 = 9
Binomial Products
A binomial expression contains two (2) terms such as : (2 + n), (2x - 5), (3n - 5p)
A binomial product is the multiplication of two such binomial expression:
(2 + n)(2x - 5), (a - b) (a + b), (2n - 5)(3n + 2), (3p + 4)2.
The product of two binomials can be obtained using two approaches:
a) the Geometrical Approach - using area diagrams.
b) the Algebraic Approach - using the distributive law.

3. Example: Expand (a + 4) ( a + 2) Algebraic Method
By the distributive law:
Geometrical Method
L = last terms
I = inside terms
(a + 4)(a + 2), can be represented by the area of a rectangle with
sides (a + 4) and (a + 2).
Area of the rectangle = length x breadth
(a + 4)(a + 2) = a a
+ 2 ) (
+ 4 a
+ 2 ( )
O = outside terms
F = first terms
= a2 + 2a + 4a + 8 a
+ 2
+ 4 ( )
Area =
= a2 + 6a + 8
The area of the large rectangle is equal to the sum of the areas of all the smaller rectangles.
F+O+I+L. method of distributive law. a
+ 4
+ 2
FOIL is a mnemonic - a memory jogger - indicating order
of multiplication (don’t forget negative signs).
F = a x a = a 2 a2
+ 4a
+ 2a
+ 8
O = a x +2 = +2 a
I = +4 x a = +4 a
L = +4 x +2 = +8
So, (a + 4)(a + 2)
= a2 + 4a + 2a + 8
= a2 + 6a + 8
(a+4)(a+2)
= a2 + 4a + 2a + 8
= a2 + 6a + 8
The algebraic method(s) are the preferred.

4. Distributive Law Method
x(5x - 2)
- 3(5x - 2)
Examples: Remove the grouping symbols:
4) (5x - 2)(x - 3) =
- - 3 x 5
-3 x 4x
= 5x2 -2x -15x + 6
1) -3(4x - 5) =
= 5x2 -17x + 6
= -12x + 15
= 3x(3x + 4)
+ 4(3x + 4)
x(x + 4)
- 7 (x + 4)
5) (3x + 4)2 =
(3x + 4)(3x + 4)
2) (x + 4) (x -7) =
= x2 + 4x - 7x - 28
= 9x2 + 12x +12x + 16
= x2 -3x - 28
= 9x2 + 24x + 16
x(2x - 1)
+ 5(2x - 1)
3) (2x - 1) (x + 5) =
Special cases : Difference of two squares.
= 2x2 - x + 10x - 5
6) (8 - x) (8 + x) =
8(8 - x)
+ x(8 - x)
= x + 8x - x2
= 2x2 + 9x - 5
= 64 - x2

5. Exercise: Remove the grouping symbols.
a) 4y(3 - 2y) =
12y - 8y
j) (2x - 1) (x + 4) + (x - 3)2
k) (2x + 3)2 - (x + 4) (x - 3)
l) (x - 3)2 + (x - 5)2 =
m) (3x - 1)2 - (4x - 1)(x + 5) =
n) (5x - 3) (5x + 3) - (4 - x) (4 + x) =
= 3x2 + x + 5
b) -8x2 (3 - 2x) =
-24x2 + 16x3
c) (x + 9)(3x + 4) =
3x2 + 31x + 36
= 3x2 + 11x + 21
d) (x - 7)(2x - 5) =
2x2 - 19x + 35
e) (3x + 2)(2x + 5) =
6x2 + 19x + 10
= 2x2 - 16x + 34
f) (x + 12)(x - 12) = x
g) (3x + 2y)2 =
9x2 + 12xy + 4y2
= 5x2 - 25x + 6
h) (7 - 2xy)2 =
xy + 4x2y2
= 26x2 - 25
i) (8x - 3y) (8x + 3y) =
64x2 - 9y2