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# Algebra

Learning Objectives ŌŚÅ To understand the following identities through geometrical proof ŌŚÅ Able to apply the identities in numerical problems. ŌŚÅ Able to recognise expressions that are factorisable. ŌŚÅ To represent inequalities in one variable, graphically.

Chapter 3

ALGEBRA Learning Objectives

ŌŚÅ 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 ŌłÆ b)2 = a2 ŌłÆ 2ab +b2 and

* (a + b)(a ŌłÆb) = a 2 ŌłÆb2 .

ŌŚÅ Able to apply the identities in numerical problems.

ŌŚÅ Able to recognise expressions that are factorisable.

ŌŚÅ 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 = ŌłÆ2 .

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

If x = ŌłÆ2 , then L.H.S = x 2 + 3x = (ŌłÆ 2)2 +3(ŌłÆ 2)

= 4 ŌłÆ 6

= ŌłÆ2 = R.H.S

Hence, this equation is true when x = ŌłÆ2.

If x = ŌłÆ1 , then L.H.S = x2 + 3x = (ŌłÆ 1)2 +3(ŌłÆ1)

=1 ŌłÆ3

ŌłÆ2 = R.H.S

Hence, this equation is true when x = ŌłÆ1.

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

= 1 +3

= 4 ŌēĀ R.H.S

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

Thus, x2+ 3x = ŌĆō2 is an equation which is true only when x takes the values ŌĆō1 and ŌĆō2. 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 ├Ś 8 =64

R.H.S = a 2 + 2ab +b2 = 32 +(2├Ś 3 ├Ś5) + 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├Ś 4 ├Ś7) + 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 ŌĆśaŌĆÖ and ŌĆśbŌĆÖ 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 Tags : Introduction to Identities | Term 3 Chapter 3 | 7th Maths , 7th Maths : Term 3 Unit 3 : Algebra
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7th Maths : Term 3 Unit 3 : Algebra : Algebra | Introduction to Identities | Term 3 Chapter 3 | 7th Maths