Algebra

Chapter 3

ALGEBRA

Learning Objectives

● To understand the following identities through geometrical proof

* (x + a )(x +b) = x ² + x(a + b) +ab

* (a + b)² =a² + 2ab +b²

* (a − b)² = a² − 2ab +b² and

* (a + b)(a −b) = a ² −b² .

● Able to apply the identities in numerical problems.

● Able to recognise expressions that are factorisable.

● To represent inequalities in one variable, graphically.

Introduction to Identities

In earlier classes, we have learnt to construct algebraic expressions using exponential notations. For example, x ² + 3x +2 is an algebraic expression in the variable x. This can also be written as an equation x² + 3x = −2 .

By substituting numerical values for x, we can verify this equation.

If x = −2 , then L.H.S = x ² + 3x = (− 2)² +3(− 2)

                                                   = 4 − 6

                                                   = −2 = R.H.S

Hence, this equation is true when x = −2.

If x = −1 , then L.H.S = x² + 3x = (− 1)² +3(−1)

 =1 −3

−2 = R.H.S

Hence, this equation is true when x = −1.

But when  x = 1 , then L.H.S = x² + 3x = (1)² +3(1)

= 1 +3

= 4 ≠ R.H.S

Thus, this equation is not true when x = 1 .

Thus, x²+ 3x = –2 is an equation which is true only when x takes the values –1 and –2. Hence, an equation is true only for certain values of the variable in it.

It is not true for all values of the variables.

Now, consider the algebraic expression, (a + b)² = a² + 2ab +b² . Let us try to find the values of the expression for the given values of a and b.

When a = 3 and b = 5,

L.H.S = (a + b)² =(3 + 5)² =8² = 8 × 8 =64

R.H.S = a ² + 2ab +b² = 3² +(2× 3 ×5) + 5² =9 + 30 +25 = 64

Thus, for a = 3 and= b = 5 , L=.H.S = R.H.S

Similarly, when a = 4 and b = 7,

L.H.S = (a + b)² =(4 + 7)² =11² =121

R.H.S = a ² + 2ab +b² = 4 ² + (2× 4 ×7) + 7² = 16 + 56 +49 =121

Also, for a = 4 and b = 7 , L.H.S = R.H.S

Thus, we shall find that for any value of ‘a’ and ‘b’ L.H.S = R.H.S. Such an equality, which is true for every value of the variable in it is called an identity. Thus, we observe that the equation (a + b)² =a² + 2ab +b² is an identity.

In general, algebraic equalities which hold true for all the values of the variables are called Identities. Let us see the basic identities with geometrical proof.

MATHEMATICS ALIVE- Inequalities in Real Life