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Algebraic Identities

Algebraic Identity Definition:

An Identity is an equality which is true for every value of the variable in it.

Example:  ( x+1 ) ( x+2 )   =    x+ 2x + x + 2
                                        =   x+ 3x + 2
         For any value of x LHS is equal to RHS,which shows the appearance of  identity here.
        Some of the identity helpful for solving the problems are given below:
  • (a + b )2 = a2 + 2ab + b2
  • (a - b )2 = a2 -2ab + b2
  • (a + b ) ( a - b ) = a2 - b2
Proof of Identity

                ( a + b)2
                 Step 1: expand the term        =  ( a + b) ( a + b)
                 Step 2: factories                 =  a ( a + b) + b ( a + b)
                 Step 3: simplify =  a+ ab + ba + b2
                 Step 4: add the common term  =  a+ 2ab + b2
                             assign a = 1, b = 2
                            (a + b)2  = a+ 2ab + b2
                            (1 + 2)2  = 12 + 2*1*2 + 22
                                (3)2    =  1 + 4 + 4
                                     9  = 9
                                LHS  = RHS

List of algebraic identity :

The following are some of the important algebraic identities or expression used in class 9th maths

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

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

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

                   4.  ( x + a ) ( x + b ) = x2 + ( a + b) x + ab

                   5.  (x + a ) ( x - b)    = x2 + ( a -b ) x - ab

                   6.  ( x -a ) ( x + b )  = x2 +  ( b - a ) x - ab

                   7.  ( x - a ) ( x - b )  = x2 -  ( a + b ) x + ab

                   8.  ( a + b )3 =  a3 + b3  + 3ab ( a + b )

                   9.  ( a - b )3  = a3  - b3 - 3ab (a - b ) 

                 10.  (x + y + z) = x2 + y2 + z2 + 2xy +2yz + 2xz

                 11.  (x + y - z)2  =  x2 + y2 + z2 + 2xy - 2yz - 2xz

                 12. ( x - y + z)2  = x2 + y2 + z2 - 2xy - 2yz + 2xz

                 13.  (x - y - z)2  = x2  + y2 + z2 - 2xy + 2yz - 2xz

                 14.  x3  + y3 + z3 - 3xyz =  (x + y + z ) ( x2 + y2 + z2 - xy - yz -xz)

                  15. x+ y2  = $\frac{1}{2}$  [( x + y)2 +  ( x - y)2

                  16. ( x + a)  ( x + b)  ( x + c)  =  x+ (a + b + c) x2 +  ( ab + bc + ca ) 
                         x +  abc

                  17.  x3 + y3  =  (x + y) ( x-xy + y)

                  18.  x3  - y =  ( x - y)  ( x+ xy + y)

                  19.  x+ y+ z-xy - yz - zx = $\frac{1}{2}$ [( x - y)+ (y -z)+ ( z - x)2]  

Examples based on Identity:

                 Example 1:  Factorize the term  (xy)2 – 82

                 Solution:     Given (xy)2 – 82
                 Step 1: First check and make use of identity (a + b) (a - b) = a2 -b2              
                 Step 2: Assign the value in identity 
                                   ( xy )2 – 82    =  ( xy + 8 ) ( xy - 8 )

                Example 2:  Expand ( x - 2y)2
         
                Solution :
                Step 1: Make use of the identity       ( a -b )2  =  a- 2ab + b2
                Step 2: Assign the value in equation  a = x, b = 2y   =  ( x)- 2 * x * 2y + (2y)2
                Step 3:   Simplify    =  x- 4xy + 4y2

Best Results From Wikipedia Yahoo Answers Youtube


From Wikipedia

Identity (mathematics)

In mathematics, the term identity has several different important meanings:

  • An identity is a relation which is tautologically true. This is usually taken to mean something that is true by definition, either directly by the definition, or as a consequence of it. For example, algebraically, this occurs if an equation is satisfied for all values of the involved variables. Definitions are often indicated by the 'triple bar' symbol ≡, such as A2≡ x·x. The symbol ≡ can also be used with other meanings, but these can usually be interpreted in some way as a definition, or something which is otherwisetautologically true (for example, a congruence relation).
  • In algebra, an identity or identity elementof a set S with abinary operation· is an element e that, when combined with any element x of S, produces that same x. That is, for all x in S. An example of this is the identity matrix.
  • The identity functionfrom a set S to itself, often denoted \mathrm{id} or \mathrm{id}_S, is the function which maps every element to itself. In other words, \mathrm{id}(x) = x for all x in S. This function serves as the identity element in the set of all functions from S to itself with respect to function composition.

Examples

Identity relation

A common example of the first meaning is the trigonometric identity

\sin ^2 \theta + \cos ^2 \theta = 1\,

which is true for all complex values of \theta (since the complex numbers \Bbb{C} are the domain of sin and cos), as opposed to

\cos \theta = 1,\,

which is true only for some values of \theta, not all. For example, the latter equation is true when \theta = 0,\, false when \theta = 2\,.

See also list of mathematical identities.

Identity element

The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all a\in\Bbb{R},

0 + a = a,\,
a + 0 = a,\, and
0 + 0 = 0.\,

Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all a\in\Bbb{R},

1 \times a = a,\,
a \times 1 = a,\, and
1 \times 1 = 1.\,

Identity function

A common example of an identity function is the identity permutation, which sends each element of the set \{ 1, 2, \ldots, n \} to itself or \{a_1,a_2, \ldots, a_n \} to itself in natural order.

Comparison

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the group of permutations of \{ 1, 2, \ldots, n \} under composition.



From Yahoo Answers

Question:Heloooooooooo, what is an algebra identity and how do i solve one. Here is one I can across on a non-calculator algebra exam paper: x^2 - ax + 144 = (the equals has 3 horizontal lines) (x-b)^2 I dont want the solution I need to understand what is going on! Thanks in advanced! how do u get the 12???? i know its the square rot of 144, but...

Answers:When the identity symbol of three lines is used it means that the two sides are equal for every value of the variable. Another way of expressing this is that there must be the same number of x's on both sides, the same number of x^2's etc. So x^2 - ax + 144 = (x - b)^2 = x^2 - 2bx + b^2 The x^2's are already equal but also 144 = b^2 ----> b = 12 or -12 -ax = -2bx ----> a = 2b So either x^2 - 24x + 144 = (x - 12)^2 or x^2 + 24x + 144 = (x + 12)^2 Edit. b^2 = 144 ----> b = +/-sqrt(144) = +/-12 As you said, it's because 12 is square root of 144. That's all there is to it!

Question:x+1/x=4 Solve x^3+1/x^3.

Answers:(x + 1)/x = 4 x + 1 = 4x x - 4x = -1 -3x = -1 x = -1/-3 x = 1/3 (x^3 + 1)/x^3 = x^3/x^3 + 1/x^3 = 1 + 1/(1/3)^3 = 1 + 1(3/1)^3 = 1 + 27 = 28

Question:I want to know what are the different algebraic identities and what are the toughest and longest identities ?? it will be helpful for my maths lab activities please help

Answers:Algebraic identities are just algebraic expressions we know are equivalent. For example: x + x = 2x is true in any algebraic structure that contains whole numbers and includes the distributivity law for multiplication and addition. That's an identity, because it doesn't matter what x is, it's always true. You can make algebraic identities very long if you want to, but generally, length of an identity is inversely proportional to how interesting and relevant it is.

Question:Represent Geometrically: (a-b)2 (a+b)(c+d) a2-b2 (a+b)2 +(a-b)2

Answers:This site answers most of these questions, and gives you the idea how to do the others. It's very difficult to explain in words. http://library.thinkquest.org/C0110248/algebra/alidentities.htm

From Youtube

MF53: Algebraic identities :Algebraic identities are at the heart of a lot of mathematics, especially geometry and analysis. Here we have a look at some simple and familiar identities, such as the difference of squares, the geometric series, and identities that go back to Pythagoras and Fibonacci. This video belongs to Wildberger's MathFoundations series, which attempts to create a coherent and logical framework for modern mathematics.

Boolean Algebraic Identities - Part 1 :What's in this Podcast? This video podcast looks at Boolean Algebraic Identies from the CCEA A2 Unit 1 section 3.7 Technology and Design specification. Keywords and Phrases: Boolean And Function Boolean Or Function Boolean Algebraic Identities - Part 1 from Podcastschool on Vimeo. Subscribe with iTunes