2. NUMBER SYSTEM
Sl
No
Type of
Numbers
Description
1 Natural
Numbers
N = { 1, 2, 3, 4, 5, . . .}
It is the counting numbers
2 Whole
Numbers
W= { 0, 1, 2, 3, 4, 5, . . .}
It is the counting numbers + zero
3 Integers Z = {. . . -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 . . .}
4 Positive
Integers
Z+ = { 1, 2, 3, 4, 5, . . . }
5 Negative
integers
Z– = {. . . -4, -3, -2, -1 }
3. Sl
No
Type of
Numbers
Description
6
Rational
Numbers
A number is called rational if it can be
expressed in the form p/q where p and
q are integers (q>0).
Ex: 4/5
7
Irrational
Numbers
A number is called rational if it cannot
be expressed in the form p/q where p
and q are integers (q> 0).
Ex: √2, Pi, … etc
8
Real
Numbers
A real number is a number that can be
found on the number line.
All rational and irrational numbers
makes the collection of Real Numbers.
[Denoted by the letter R]
4. Natural
Numbers N
1, 2, 3, . . .
Whole Numbers W
0, 1, 2, 3, . . .
Integers Z
. . ., -2, -1, 0, 1, 2, 3,
. . .
Rational Numbers Q
0 1 5 ½ -⅔ -9 Irrationals
√2
√3
𝝅
0.1011011
1011110...
REAL NUMBERS
Real Numbers
5. Sl
No
Type of
Numbers
Description
9
Real
numbers &
their
decimal
Expansions
The decimal expansion of a rational
number is either terminating or non
terminating recurring. Moreover, a
number whose decimal expansion is
terminating or non-terminating recurring
is rational.
The decimal expansion of an irrational
number is non-terminating non-
recurring. Moreover, a number whose
decimal expansion is non-terminating
non-recurring is irrational.
6. Sl
No
Type of
Numbers
Description
10
Operations
on Real
numbers
The sum or difference of a rational
number and an irrational number is
irrational
The product or quotient of a non-zero
rational number with an irrational
number is irrational.
If we add, subtract, multiply or divide
two irrationals, the result may be
rational or irrational.
18. Sl no Terms Description
1 Definition
A polynomial expression P(x) in one
variable is an algebraic expression in x
where power of the variable is whole
number and coefficients are real
numbers.
Polynomials
19. Sl no Terms Description
3 Degree
Highest power of the variable in a
polynomial is the degree of the
polynomial.
4
Terms of a
polynomial
expression
The several parts of a polynomial
separated by ‘+’ or ‘-‘ operations are
called the terms of the expression.
Ex :
20. Type of
polynomial
Degree Form
Constant 0 P(x) = a
Linear 1 P(x) = ax + b
Quadratic 2 P(x) = ax2 + ax + b
Cubic 3 P(x) = ax3 + ax2 + ax + b
Bi-quadratic 4 P(x) = ax4 + ax3 + ax2 + ax + b
#5 Types of Polynomial based on their
Degrees
21. Type of
polynomial
Degree Form
Monomial 1
Polynomials having only one term are
called monomials (‘mono’ means
‘one’).
e.g., 13x2
Binomial 2
Polynomials having only two terms are
called binomials (‘bi’ means ‘two’).
e.g., (y30 + √2)
Trinomial 3
Polynomials having only three terms
are called trinomials (‘tri’ means
‘three’).
e.g., (x4 + x3 + √2), (µ43 + µ7 + µ) and
(8y – 5xy + 9xy2) are all trinomials
#6 Types of Polynomial based on the
number of terms
22. Sl no Terms Description
7
Zeroes or
Roots of a
Polynomia
l
A zero of a polynomial p(x) is a number
c such that p(c) = 0.
If P(a) = 0, then ‘a’ is the zero of the
polynomial P(x) and the root of the
polynomial equation P(x) = 0.
Note:
● A non-zero constant polynomial
has no zero.
● By convention, every real number
is a zero of the zero polynomial.
● The maximum number of zeroes
of a polynomial is equal to its
degree.
23. Sl no Terms Description
8
Factor
Theorem
When a polynomial f (x) is divided by
(x – a), the remainder = f (a). And, if the
remainder
f (a) = 0, then (x – a) is a factor of the
polynomial f(x).
Note: We have to know factor theorem
in order to factorize cubic polynomials.
24. Sl
no
Terms Description
9
Factorization
of a
Polynomial
By taking out the common factor:
If we have to factorize x2 –x then we can
do it by taking x common.
x(x – 1) so that x and x-1 are the factors of
x2 – x.
By grouping:
ab + bc + ax + cx = (ab + bc) + (ax + cx)
= b(a + c) + x(a + c)
= (a + c)(b + x)
By splitting the middle term:
Write the given quadratic Polynomial in standard
form.
Find the Product of a and c.
List down all factors of ac in pairs.
Select a pair of factors such that their sum is ‘b’.
Now split the middle term ‘b’, in terms of the
factors found.
