The document defines algebra and its basic concepts such as variables, terms, algebraic expressions, types of algebraic expressions including monomials, binomials, trinomials, and polynomials. It also discusses operations like addition, subtraction, multiplication, and division of algebraic expressions. For addition and subtraction, terms are grouped based on their literal coefficients and numerical coefficients are added or subtracted. Multiplication involves multiplying the coefficients and variables separately. Division of monomials, polynomials by monomials, and polynomials by polynomials is explained through examples.

2. DEFINITION OF ALGEBRA:-
 Algebra is a branch of mathematics that uses letters
etc. to represent numbers and quantities.

3. ELEMENTARY TREATMENT:-
1. CONSTANTS: In Arithmetic, we use digits
(0,1,2,3,4,5,6,7,8,9)each of which has a fixed value
and so are called constants.
2. VARIABLES: We use letters of English alphabet
which can be assigned any value according to the
requirement. So the letters used in algebra are
called variables.

4. 3. TERM: A term is a number (constant),a variable or
a combination (product or quotient) of number and
variables. For e.g.:-
5, 3a, x, ax, -2xz, 4x/3y.etc.
4.ALGEBRIC EXPRESSION: An algebraic
expression is a collection of one or more terms,
which are separated from each other by addition
(+) or subtraction (-) sign(s). For e.g.:-5xy – 2 ,3x +
5z - 2xy , ax + by – cz + axy + dxyz.etc.

5. NAME
CONDITION
EXAMPLES
i. Monomia
l
has only one term X, 6xy, -4xy/9yz,
etc.
ii. Binomial has two terms 2a+x, 7z-15x, etc.
iii. Trinomial has three terms ax2+bx+c, x2-xy-
9y+x/2, etc.
iv.Multinomia
l
has more than
three terms
4-a+ax+by, x2-
2x+3xy-3y-b+ ,
etc.
v.
Polynomial
has two or more
than two terms.
Every binomial,
trinomial &
5. TYPES OF ALGEBRIC EXPRESSION:-

6. 6. PRODUCT : When two or more quantities (constants,
variables or both) are multiplied together; the result is
called their product. E.g.:- 4ay is the product of 4,a and y.
7.FACTOR: Each of the quantities (numbers or variables)
multiplied together to form a term is called a factor of the
term. E.g. :- 3,a and y are the factors of the term 3ay.
8.CO-EFFICIENT: In a monomial, any factor or a group of
factors of a term is called the coefficient of the remaining
part of the monomial. E.g. :- In 5xyz, 5 is the coefficient of
xyz, x is the coefficient of 5yz, y is the coefficient of 5xz
and so on.

7. 9. TYPES OF CO-EFFICIENTS:-
i. NUMERICAL CO-EFFICIENTS: - If a factor is
a numerical quantity , it is called numerical co-
efficient. E.g. :- In 4abc, 4 is the numerical co-
efficient.
ii. LITERAL CO-OEFFICIENT:- The factor
involving letters is called the literal co-
efficient. E.g. :- In 7xyz,
7, x , y , z , 7x , 7y , 7z , xy , yz , xz ,7xy ,7yz
,7xz ,xyz and 5xyz are the literal co-efficients.

8. 10. DEGREE OF A MONOMIAL:-The degree of a monomial is the
exponent of its variable or the sum of the exponents of its
variable. E.g. :-
i.
ii.
iii.
11.DEGREE OF A POLYNOMIAL :- The degree of a polynomial is the
degree of the highest degree term. E.g. :-

9. 12.LIKE TERMS :- Terms having the same literal coefficients or
alphabetic letters are called like terms. Like terms can be
added, subtracted, multiplied and divided. E.g.:-
i.
ii.
iii.

10. 13.UNLIKETERMS:- Terms having different literal coefficients
are called unlike terms. Unlike terms cannot be added,
subtracted, multiplied or divided. E.g.:-
i.
ii.
iii.

11. ADDITION & SUBTRACTION
( Like Terms )
 Method:- Add or subtract (as required) the
numerical coefficients of like terms. E.g. :-
i. Addition of 8xy, 15xy and 3xy = 8xy + 15xy + 3xy
= ( 8 + 15 + 3 ) xy
= 26xy ( Ans )
ii. Subtraction of 8xy from 15xy = 15xy – 8xy
= ( 15 – 8 ) xy
= 7xy ( Ans )

12.  Example 1 :
Add: (i) 2ax + 3by + 4cy, 5by – 3cy – ax and 6cy +
4ax – 9by.
Solution:
1st method : Re-write the given expression in such a
way that their like terms are one below the other,
then operate (add or subtract, as the case may be)
like terms column-wise. Thus:
2ax + 3by + 4cy
- ax + 5by – 3cy
4ax – 9by + 6cy
5ax – by + 7cy (Ans.)

13.  2nd method :
Steps :
i. Write each of the given polynomials (expressions)
in a bracket with plus sign(+) between the
consecutive brackets.
ii. Remove the brackets without changing the sign
of any term.
iii. Group the like terms and add.
E.g.:- ( 2ab + 3yz + 4xz ) + ( 5yz – 3xz –ab ) + ( 6xz +
4ab – 9yz )
= 2ab + 3yz + 4xz + 5yz – 3xz –ab + 6xz + 4ab
– 9yz
= 2ab – ab + 4ab + 3yz + 5yz – 9yz + 4xz – 3xz

14.  Subtract: (i) 2a + 3b – c from 4a + 5b + 6c.
Solution:
1st method:
i. Write the given expressions in two rows in such a
way that the like terms are written one below the
other, taking care that the expression to be
subtracted is written in the second row.
ii. Change the sign of each term in the second row
(lower row).
iii. With these new signs of the terms in lower row,
add the like terms column wise.
Step 1: 4a + 5b + 6c
2a + 3b - c
Step 2: - - + .

15.  2nd method:
i. Write both the expressions in a single row with the
expression to be subtracted in a bracket and put a
minus sign (-) before(outside) this bracket (minus
sign is for subtraction).
ii. Open the bracket by changing the sign of each
term inside the bracket.
iii. Add the like terms.
E.g. 4a + 5b + 6c – ( 2a + 3b – c )
[ Step 1 ]
= 4a + 5b + 6c – 2a – 3b + c
[Step 2 ]
= 4a - 2a + 5b – 3b + 6c + c

16. ADDITION & SUBTRACTION
( Unlike Terms )
 The two like terms can be added or subtracted to get
a single term; but two unlike terms cannot be added
or subtracted together to get a single term. All that
can be done is to connect them by the sign as
required.
 E.g. :
i. Addition of 5x and 7y = 5x + 7y
ii. Subtraction of 9y from 3xy = 3xy – 9y

17. MULTIPLICATION
 Multiplication of monomials :
Steps : 1. Multiply the numerical coefficients
together.
2. Multiply the literal coefficients separately
together.
E.g. :
i. 8x x 3y = ( 8 x 3 ) x ( x x y )
= 24 x xy
= 24 xy
ii. 5a x 3b x 6c = ( 5 x 3 x 6 ) x ( a x b x c )
= 90 abc

18.  Multiplication of a polynomial by a monomial :
Steps: 1. Write the given polynomial inside the
brackets and the monomial outside it.
2. Multiply the monomial with each term of
the polynomial and simplify.
E.g. :
i. Multiplication of 2x + y – 8 and 4x = 4x ( 2x + y – 8
)
= 4x x 2x + 4x
x y – 4x x 8
=
(Ans.)

19.  Multiplication of a polynomial by a polynomial :
Steps: 1. Multiply each term of one polynomial
by each term of other polynomial.
2. Combine (add or subtract) the like
terms.
E.g.:
Multiplication of a + b and 2a + 3b,
a + b
2a + 3b
[Multiplying
first polynomial by 2a]
………………………………………
[Multiplying first polynomial by 3b]
………………………………………

20.  Alternative method:
Multiplication of a + b and 2a + 3b = (a + b)
(2a + 3b)
= a(2a +
3b) + b(2a + 3b)
= a(2a) +
a (3b) + b (2a) + (3b)
=
=
.

21. DIVISION
 Dividing a monomial by a monomial :
Steps: 1.Write the dividend in numerator and
divisor in denominator.
2. Simplify the fraction, obtained in Step
(1).
E.g.
i. Division of 15xy by 5x = = 3y .
ii. .
iii. .

22.  Dividing a polynomial by a monomial :
Divide each term of the polynomial by the
monomial:
E.g. :
i. Division of
.
ii.

23.  Dividing a polynomial by a polynomial:
E.g.:
Divide: by
Step 1: Set the two expressions as:
Step 2: Divide first term of the dividend by the first term
of divisor to get
the first term of quotient. Here, ;
which is the first
term of the quotient .

24.  Step 3: Multiply quotient (2x) with each term of
divisor and write as :
 Step 4: Subtract the result of Step 3 and take the
next term/terms of the dividend down :

25.  Step 5 : Repeat the process from Step 2 to Step 4
taking remainder
15x + 10 as new dividend :
Thus,
(Ans.)