The document explains the fundamentals of algebra, detailing algebraic expressions, constants, variables, and the types of algebraic expressions including monomials, binomials, and polynomials. It covers operations such as addition, subtraction, and multiplication of algebraic expressions, as well as standard identities useful in simplifying expressions. The document also provides rules and examples to illustrate how to perform these operations effectively.

1. ALGEBRAIC
EXPRESSIONS AND
IDENTITIES

2. WHAT IS ALGEBRA?
• Algebra is one of the broad parts of
mathematics, together with number theory,
geometry and analysis. In algebra, we come
across two types of symbols, constants and
variables.
• CONSTANT- a symbol having a fixed numerical
value. A constant term is called
• VARIABLE- a symbol having various numerical
value

3. WHAT ARE ALGEBRAIC
EXPRESSIONS?
• The combination of constants and variables
connected by signs of fundamental operations of
addition ,subtraction ,multiplication and division.
Ex- 2x²+ 3xy-4y²
• Terms- 2x², 3xy, 4y²
• Factors- 2x²= 2* x *x. 3xy= 3*x*y. 4y²= 4*y*y. A
constant term is called numerical factor and a
variable term is called literal factor.
• Coefficients- There are 2 types of coefficients-
Numerical and Variable
• Coefficient of x² in 2x²- 2 (Numerical)
• Coefficient of 3 in 3xy- xy (Variable)

4. Algebraic expressions consist of polynomials which
have monomials, binomials, trinomials and other
terms.
• Monomials- single term in an expression.
• Binomials- two terms in an expression.
• Trinomials-Three terms in an expression.
• Polynomials- any no. of terms in an expression.
TYPES OF ALGEBRAIC EXPRESSIONS-:

5. LIKE AND UNLIKE TERMS
• LIKE TERMS- Terms having same literal
factor.
• UNLIKE TERMS- Terms not having same
literal factors.
• Ex- x² and y² (unlike terms)
• 4xy and -2xy (like terms)
• x² and -x² (like terms)

6. ADDITION AND SUBTRACTION
OF ALGEBRAIC EXPRESSIONS
• Add 7x+y²+7xy and 4x+3xy
= 7x+y²+7xy+4x+3xy
= 7x+4x+y²+7xy+3xy
= 11x+y²+10xy
• Subtract 6xy-24a+10x and 5x-4a+7xy
= 6xy -24a +10x – 5x-4a+7xy
= (6xy +7xy)-(24a -4a) +(10x -5x)
= 13xy-20a+5x

7. MULTIPLICATION OF
ALGEBRAIC EXPRESSIONS
In multiplication of algebraic expressions, we shall be
using the following rules of signs:-
1) The product of 2 factors with like signs is positive
and the product of 2 factors with unlike signs is
negative i.e
• (+)*(+)= (+)
• (+)*(-)= (-)
• (-)*(-)= (+)
• (-)*(+)= (-)
2) If a is any variable and m , n are positive integers,
then
aᵐ*aᵑ= aᵐᶧᵑ

8. WHAT CAN WE MULTIPLY?
We can multiply a :-
• Monomial with a monomial
• Monomial with a binomial
• Monomial with a trinomial
• Binomial with a binomial
• Binomial with a trinomial
• Trinomial with a trinomial

9. MONOMIAL*MONOMIAL
• We have, (4ab)*(5a) so we will first simply
multiply the constants or the numerical
values.
• After that we will multiply the variables.
• (4a)*(5a)
= 4*5*(a*a)
= 20*a²
= 20a²

10. • 4*5= 20 ,
Coefficient of product = coefficient of first
monomial*coefficient of second monomial.
• a*a= a²
Algebraic factor of product = algebraic factor of
first monomial*algebraic factor of second
monomial.
RULES

11. MONOMIAL*BINOMIAL
• To multiply a monomial by a binomial we
have to use the distributive property.
• Ex- 6x*(3x+4y) (Using the distributive property)
= 6x*3x+6x*4y
= 18x²+24xy

12. MONOMIAL*TRINOMIAL
• In the multiplication of a monomial by a trinomial
we multiply each term of the trinomial by the
monomial and add the products.
• Ex- 12x*(4x+2y+5z)
= 12x*4x +12x*2y +12x*5z
= 48x² +24xy +60xz

13. BINOMIAL*BINOMIAL
•In the multiplication of 2 binomials we have to
use the distributive property of multiplication of
literals over their addition.
• Ex- (3a+4b)*(2a+3b)
= 3a*(2a+3b) + 4b*(2a+3b)
= 3a*2a +3a*3b +4b*2a +4b*3b
= 6a² +9ab +8ab +12b²
= 6a²+17ab+12b²
• It follows from above example that to multiply
any 2 binomials, we multiply each term of 1
binomial by each term of the other and add the
products

14. BINOMIAL*TRINOMIAL
TRINOMIAL*TRINOMIAL
• To multiply a binomial and a trinomial or a
trinomial and a trinomial we will use the same
steps as used before i.e. Using the distributive
property and multiplying each of the 3 terms in the
trinomial by each of the 2 or 3 terms in the
binomial and trinomial.

15. STANDARD IDENTITIES
• IDENTITY- It is an equality which is true for
all values of the variable (s).
• Standard identities are useful in carrying
out squares and products of algebraic
expressions and they also allow easy
alternative methods to calculate numbers
of products.

16. STANDARD IDENTITIES
• The 3 standard identities are :-
1. (a + b)² = a² + 2ab + b²
2. (a - b)² = a² - 2ab + b²
3. (a + b) (a - b) = a² - b²
• There are 3 standard identities but there is
another useful identity :-
4. (x + a) (x + b) = x² + (a + b) x + ab

17. • THEOREM-
(a + b)² = (a + b) (a + b)
= a(a + b) + b(a + b)
= a² + ab + ba + b²
= a² + 2ab + b² (since ab=ba)
∴ (a + b)² = a² + 2ab + b²
IDENTITY 1- (a + b)² = a² + 2ab + b²

18. IDENTITY 2- (a - b)² = a² - 2ab + b²
• THEOREM-
(a - b)² = (a – b) (a – b)
= a(a – b) - b(a - b)
= a² - ab – ba + b²
= a² - ab – ab + b² (sinceab=ba)
= a² - 2ab + b²
∴ (a - b)² = a² - 2ab + b²

19. Identity 3 - (a + b) (a - b) = a² - b²
• THEOREM-
(a + b) (a - b) = a(a - b) – b(a – b)
= a² - ab + ba + b²
= a² - ab + ab + b² (since ba = ab)
= a² - 2ab + b²
∴ (a + b) (a - b) = a² - b²

20. IDENTITY 4- (x+a)(x+b)=x²+(a+b)x+ab
• THEOREM –
x(x+b)+a(x+b)=x²+bx+ax+ab
=x²+x(b+a)+ab
or
= x²+x(a+b)+ab
∴(x + a) (x + b) = x² + (a + b) x + ab