In arithmetic we express 1+ 1+ 1= 3, 2 x 2 x
2 = 8, 3 – 2 = 1 etc., Here 1, 2, 3, 8 are known numbers. In algebra we deal
with symbols like x, y, z, a, b, c, p etc., which represent unknown numbers. .
x + x + x = 3x, a x a x a = a3,
3x - 2x = x are some examples of algebraic statements. In the above examples 1,
2, 3 and 8 are known and x, y, z, c, b, c and p are unknowns. .
When x + y = 10, the value of x depends on
the value of y and vice-versa. Here 10 is a constant and x, y are variables. .
We can define algebra as a branch of
mathematics which deals with rules, procedures and operations involving wns and
unknowns; and constants and variables.
Algebraic expressions are formed from
variables and constants. 3x + 4y is an algebraic expression. The value of an
expression depends on the values of the variables from which the expression is
Formulas in algebra are written in a general
form using algebraic expressions. .
The following algebraic formulas should be
memorized since they are very useful to solve certain problems. .
(a + b)2 = a2 + 2ab +
(a - b)2 = a2 - 2ab +
(a + b)2 = a2 + b2
+ c2 + 2ab + 2bc + 2ca
(a + b)3 = a3 + b3
+ 3ab (a + b)
(a - b)3 = a3 – b3
- 3ab (a - b)
a3 + b3 = (a + b) (a2
- ab + b2)
a3 - b3 = (a - b) (a2
+ ab + b2)
a3 + b3 + c3
- 3abc = (a + b + c) (a2 + b2 + c2 - ab - bc -
if a + b + c = 0, then a3 + b3
+ c3 = 3abc. .
Worked out examples
If a = 13, b = 12, then find the value of
Given that a = 13, b = 12. .
= a + b
= = a + b
= 13 + 12 = 25
x = 103, then find the value of
(\ a3 - 3ba2 + 3ab2 – b3
= (a - b)3)
= x - 2 = 101
If y + = 10,
then find the value of .
y + = 10
Multiplying by y on both sides,
y2 + 25 = 10y
y2 - 10y + 25 = 0
(y - 5)2 = 0
y - 5 = 0
y = 5
If 3y + = 3,
then find the value of 64y3 + .
3y + = 3
Multiplying both sides by 4y + = 4.
Cubing on both sides,
64y3 + + 3(4y)= 64
(\ (a + b)3 = a3 + b3 + 3ab (a +
64y3 + + 4 (4)
64y3 + = 48
If y + , then
find the value of