MATHS7/10:FUNCTIONAL & CONDITIONAL IDENTITIES : MATHS7/10:FUNCTIONAL & CONDITIONAL IDENTITIES BY
B. MANJULA
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FUNCTIONS : FUNCTIONS In an algebraic expression, x 2 – x – 2, its value depends upon value of x, which is a variable. It is expressed as f(x) = x2 – x – 2
Kinds of functions:
(a)Linear function : f(x) = 2x + 3
(b)Quadratic function :f(x) = x2 – 3x +1
© Polynomial function:f(x) = 2x4 – x3 +2x2 -3x + 1
These are also called first degree, second degree and 4th degree functions respectively.
CONDITIONAL IDENTITIES : CONDITIONAL IDENTITIES A Statement which is true for all values of the unknown is called an identity
The identities to be printed by using the conditions are called conditional identities
Ex: if (a+b+c) = 0, then a2 + b2+ c2 = 3abc
If (a+b+c) = 0, b2 -4ac is a perfect square
All equations are not identities x2+y2=a2 is not an identity
IDENTITIES : IDENTITIES Consider (1) a2 – 2ab +b2 and b2 -2ba + a2
(2)a2(b-c)+ b2(c-a) + C2 (a-b)
In the first case, no change in value even when a and b are interchanged and this is called absolute symmetry
In the second case, abc are interchanged a by b , b by c, c by a and is expressed as a sum.
This is called a cyclic symmetry given by notation zigma (?) and written as ? a2(b-c) or ?b2(c-a) or ? c2(a-b)
IDENTITIES IN MATHEMATICS : IDENTITIES IN MATHEMATICS 1)(a+b)2 = a2 +2ab +b2
2)(a-b)2 = a2 – 2ab +b2
3)(a+b)(a-b) = a2 - b2
4)(a + b + c)2 = a2 + b2 +c2 +2ab + 2bc + 2ca
5)(x+a) (x + b) = x2 + x (a + b) + ab
6)(x+a)(x+b)(x+c) = x3 + x2(a + b + c)+x(ab + bc + ca) + abc
IDENTITIES IN MATHEMATICS-CONTD : IDENTITIES IN MATHEMATICS-CONTD 7)(a+b)3 = a3 + b3 +3ab(a + b)
8)(a-b)3 = a3 – b3 -3ab(a + b)
9)a3 + b3 = (a + b) (a2 – ab + b2)
10)a3 – b3 = (a – b)(a2+ab + b2)
11)a3 + b3+c3=-3abc = (a+b+c)(a2+b2+c2-ab-bc-ca)
12)(a-b)(b-c)(a-c) = ab(a-b)+bc(b-c)+ca(c-a)
13)a4 + a2b2 + b4= (a2 +ab +b2)(a2-ab+b2)
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