Algebra :

Introduces the concept of variables, which allows an unknown quantity to be represented by a alphabets to manipulation in applications.

Algebraic expression: an expression which has variable ( alphabets), constant , operations ( addition ,subtraction,multiplication division )and exponents. We try to understand relation between two different values of a statement .

For example

3 times of x means ( statement ) 3x( expression)

45 increased by some 6x ……… 45+6 x

30 decreased by some 2x ………30-2x

Types Algebraic expressions:

Monomial , binomial , trinomial and multinomial expressions

Monomial Expression: An Algebraic expression with single term .

Example: 2x+3x …..x is only variable(addition with monomial with monomial)

                4xy - 3xy.....x and y are variables(subtraction with monomial with monomial )

                 (3xy)(2x+5x) ......x and y are variables ( multiplication with monomial with binomial)

Binomial Expression: An algebraic expression with two terms ,

Examples : (5x+6y) +4x ……x and y are variables(addition binomial with monomial )

(3a - 2b ) - (2a-2b)……a and b are variables. (subtraction with binomial with binomial)

b³/2 * c/3……..b and c are variables.

Trinomial Expression : An algebraic expression with three terms,

Examples: (3x+2y-7xy) +(2x+3y+5xy)…..X and Y are variables(addition)

( 2a+ 5a + 7) - (a+3a+4) …… a is the only variable(subtractions)

(xy)(x+y+z) ………..x ,y and z are variables (multiplication)

Multinomials: An algebraic expression with one ,two or more than terms.

Examples:
m + 5mn – 7mn + nm

3 + 5x- 4xy + 5xy

Polynomials:

An algebraic expression, in which variable(s) does (do) not occur in the denominator,

exponents of variable(s) are whole numbers and numerical coefficients of various

terms are real numbers, is called a polynomial.

Types of Polynomials on the basis of degree:

Linear Polynomial: A polynomial of degree 1 is called a linear polynomial.Linear equations contains variable and constant , variables (x , y etc )does not have Exponents, square roots,and cube roots.

Example :2(x - 1) + 3(7x) - 10

Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomial.

Quadratic Equation: aX² + bx + c = 0

Example: ax² + bx + c

Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial.

Bi-quadratic Polynomial: A polynomial of degree 4 is called a biquadratic polynomial

Algebraic equations : An algebraic expression is separated by equal to( = ), that means left hand side and right hand side. Algebra equation contains the terms like numbers, integers, fractions, roots, exponents etc. Linear equations and quadratic equations are the example of the algebraic equations.

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

Algebraic Expressions and Equations and Identities

Algebraic Identities