REAL NUMBERS
EUCID'S DIVISION ALGORITHM
Given two positive integers a and b, there exist unique pair of whole numbers q and r satisfying a=bq+r, 0≤ r < b
FUNDAMENTAL THEOREM OF ARITHMATIC
Every composite number can be expressed (factorised) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur.
1. x be a rational number whose decimal form is terminating decimal. Then x can be expressed in the form of p/q , where p and q are coprimes, and the prime factorization of q is in the form 2nx 5m (where n, m are non-negative integer)
2. x = p/q is a rational number, such that the prime factorization of q is in the form 2nx 5m. (where n and m are non-negative integers) Then, x has terminating decimal.
3. x = p/q is a rational number, such that the prime factorization of q is not in the form 2nx 5m. (where n and m are non-negative integers) Then, x has a non-terminating and repeating (recurring) decimal.
DEFINITION OF LOGARITHM
a, N are two positive real numbers and a≠1, we define log a N = x⟺ax = N
LAWS OF LOGARITHM
Product Rule:
log a xy = log a x + log a y
Quotient Rule:
log a (x/y) = log a x - log a y
Power Rule:
log a xn = n log a x
SETS
Definition of a Set
Roster form of a Set and Set builder form of a Set
Belongs to and Does not belongs to
Empty set / Null set / Void set
Universal Set and Subset
Equal Sets
Finite Set and Infinite Set
Cardinality of a Finite Set
Union of Sets
Intersection of Sets
Disjoint Sets( A⋂B = ∅ )
Difference of Sets
Venn Diagrams
Formulae: n(A) + n(B) = n(AUB) + n(A⋂B)
(AUB) - (A⋂B) = (A-B) U (B-A)
POLYNOMIAL
Definition of Polynomials
Types of Polynomials:
According to number of terms: Monomial, Binomial, Trinomial and Multinomial
According to degree: Zeroth Polynomial, Constant Polynomial, Linear Polynomial,
Quadratic Polynomial, Cubic Polynomial and nth degree Polynomial
Value of a Polynomial
Zeroes of a Polynomial:
Linear polynomial has exactly one zero.
Quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero) or no zero. A quadratic polynomial has atmost two zeroes.
Cubic polynomial has atmost three zeroes.
Graph of a Polynomial:
The graph of Linear polynomial is a straight line.
The graph of Quadratic polynomial is Parabola.
The graph of Cubic polynomial is a Curve.
Zeroes are the x-coordinates of the points where the graph of Polynomial intersects the X-axis.
Relationship between zeroes and coefficients of a Polynomial:
COORDINATE GEOMETRY