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 ) = x²+ 2x + x + 2

= x²+ 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 = a²+ ab + ba + b²

Step 4: add the common term = a²+ 2ab + b²

assign a = 4, b =6

(a + b)² = a²+ 2ab + b²

(4 + 6)² = 16 + 2*4*6 +36

(10)² = 16 + 48 +36

100 = 100

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

^Derivation

(a+b)² = (a+b) (a+b)

factories = a(a+b) + b(a+b)

= a*a + a*b + b*a + b*b

= a² +2ab+ b²

2. ( a - b)² = a² - 2ab + b²

^Derivation

(a-b)² =(a-b)(a-b)

factories = a(a-b)- b(a-b)

= a*a -a*b - b*a +(- b)*(-b)

= a²-2ab+ b²

3. a² - b^{2 =}(a+b)(a-b)

^{^Derivation}

a² - b^{2 =}(a+b)(a-b)

factories = a(a-b)+b(a-b)

= a² –ab+ab-b²

= a²- b²

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

Derivation

( x + a)(x +b) = x(x+b)+a(x+b)

factories =x*x + xb + ax + a*b

=x²+(a+b)x+ab

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

derivation

( x + a)(x - b) = x(x-b) + a( x-b)

factories =x*x - xb + ax - a*b

=x²+(a - b) x- ab

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

derivation

( x - a)(x + b) = x(x+b) - a( x+b)

factories =x*x + xb - ax -a*b

=x²+(b - a) x- ab

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

derivation

( x - a)(x - b) = x(x-b) - a( x-b)

factories = x*x - xb - ax +a*b

=x²-( a+ b) x + ab

8. ( a + b )³ = a³ + b³ + 3ab ( a + b )
Derivation
(a+b)(a+b)²=(a + b)(a² + 2ab + b²)

                                    =a(a² + 2ab + b²)+ b(a² +2ab+b²)
                                     =a³ + 2(a²)b+ab² + (a²)b+2ab² + b³
                                               =a³ + 3(a²)b+3a(b²) + b³

= a³ + b³ + 3ab ( a + b )

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

(a-b)(a-b)²=(a-b)(a²-2ab+b²)

                                       =a(a²-2ab+b²)-b(a²-2ab+b²)
                                       =a³-2(a²)b+ab²-(a²)b+2ab²–b³
                                       =a³-3(a²)b+3a(b²)-b³

= a³ - b³ - 3ab ( a b )

10. (x + y + z)² = 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. x²+ y² = 12 [( x + y)² + ( x - y)²]

16. ( x + a) ( x + b) ( x + c) = x³+ (a + b + c) x² + ( ab + bc + ca )

x + abc

17. x³ + y³ = (x + y) ( x²-xy + y²)

18. x³ - y³ = ( x - y) ( x²+ xy + y²)

19. x²+ y²+ z²-xy - yz - zx = 12 [( x - y)²+ (y -z)²+ ( z - x)²]

Examples based on Identity:

Example 1: Factorize the term x² – 9²

Solution: Given x² – 9²

Step 1: First check and make use of identity (a + b) (a - b) = a² -b²

Step 2: Assign the value in identity x ² – 9 = ( x + 9) ( x - 9 )

Example 2: Expand ( x - 4y)²

Solution :

Step 1: Make use of the identity ( a -b )² = a²- 2ab + b²

Step 2: Assign the value in equation[ a = x, b = 4y ] = ( x)²- 2 * x * 4y + (4y)²

Step 3: Simplify = x²- 8xy +164y²

Entrance Exams

Thursday, July 2, 2015

Algebraic expressions , algebraic eequations and identities

Algebraic Identities