Algebraic Manipulation

 

Contents

 

Simple Algebraic Expressions


Evaluation of Algebraic Expression

 

Rules of Algebra

 

Algebraic Fractions

 

Quiz 1

 

Expansion: The Foil Method

 

Algebraic Identities

 

Basic Factorisation

 

Factorisation of Quadratic Polynomials

 

Factorisation by Grouping

 

Quiz 2

 

Long Division

Polynomial Identities

Quiz 3

Algebra Main Page

 

 

Geometric Representation of Algebraic Identities

Very often, algebraic manipulation can be explained with the aid of geometrical figures. 

Activity 1: 

Objective: To give a demonstration of (a+b)2 = a2 +2ab +b2

1. The figure on the right is a square made up of four parts A, B, C & D. The sides of the square are each (a+b) units. Therefore the area of the square is (a+b) (a+b) or (a+b)2 square units.

a) Is the area of A equals to a2 square units?

b) Is the area of B equals to ab square units?

c) Is the area of C equals to ab square units?

d) Is the area of D equals to b2 square units?

Is (a+b)2 = a2 +2ab +b2

 
Activity  2:
 
Objective: To give a demonstration of (a-b)2 = a2 -2ab +b2 
 
2) The figure on the right is made up of four parts A, B, C & D. A, B & C made up a square, which has an area of a2 square units. The area of D is b2 square units. 
 
a)Is the area of A equals to (a-b)2 square units?

b) Is the area of B equals to ab square units?

c) Is the area of rectangle made up of C &D equals to ab square units?

d) Is a2 + b 2 = (a-b)2 +ab+ ab true?

e) Is (a-b)2=a2 -2ab +b2 true?

 
Activity 3:
 
Objective: To give a demonstration of (a+b) (a-b) = a2 -b2
 
A small square, B, of area b2 square units is removed from a bigger square of area a2 square units as shown in the figure on the right. The area of the remaining figure, A, is (a2 - b2) square units.
 
Suppose part A is cut into two portions and rearranged to form a rectangle as shown.
 
a) Is the area of the newly formed rectangle equal to (a+b) (a-b) square units?
 
b) Is a2 -b2 =(a+b) (a-b) true?
 
Note that all the answers to the above questions are "yes".
 
The above activities prove the three algebraic identities which will also hold true:
 

(a+b)2 = a2 +2ab +b2

(a-b)2 = a2 -2ab +b2

(a+b) (a-b) = a2 -b2

         

Example 1:

(3000 + 1)2  

    = 9 000 000 + 6 000 +1

    = 9 006 001
 

Example 2:
 
(2a3b - 3)2
 
    = 4a6b2 - 12a3b +9
 

Example 3:
 
(-3b - 2a ) (-3b + 2a)
 
     = 9b2 - 4a2