2] Oswaal CBSE Chapterwise & Topicwise Revision Notes for Term-I, Mathematics (Basic), Class-X
p
Non-terminating Repeating (or Recurring) Decimals: The decimal expansion obtained from repeats
q
periodically, then it is called non-terminating repeating (or recurring) decimal.
Just divide the numerator by the denominator of a fraction. If you end up with a remainder of 0, you have a
terminating decimal otherwise repeating or recurring decimal.
The sum or difference of a rational and an irrational number is irrational.
The product and quotient of a non-zero rational and an irrational number is irrational.
p
Let x = be a rational number, such that the prime factorization of q is of the form 2m5n, where n and m are non-
q
negative integers. Then, x has a decimal expansion which terminates.
p
Let x be a rational number whose decimal expansion terminates. Then, x can be expressed in the form , where p
q
and q are co-primes and the prime factorization of q is of the form 2m5n, where m and n are non-negative integers.
p
Let x = be a rational number, such that the prime factorization of q is not of the form 2m5n, where n and m are
q
non-negative integers. Then, x has a decimal expansion which is non-terminating repeating.
Know the Formulae
For two positive integers a and b, we have
HCF (a, b) × LCM (a, b) = a × b
a×b
or HCF (a, b) =
LCM ( a , b )
a×b
and LCM (a, b) =
HCF ( a , b )
qq
UNIT II: ALGEBRA
Chapter - 2 : Polynomials
Revision Notes
Zeroes of a Polynomial and Coefficients of
Quadratic Polynomials
Polynomial: An algebraic expression in the form of anxn + an–1xn–1 +.........+ a2x2+a1x + a0, (where n is a whole
number and a0, a1, a2, ........., an are real numbers) is called a polynomial in one variable x of degree n.
Value of a Polynomial at a given point : If p(x) is a polynomial in x and ‘α’ is any real number, then the value
obtained by putting x = α in p(x),
is called the value of p(x) at x = α.
Zero of a Polynomial: A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.
Geometrically, the zeroes of a polynomial p(x) are precisely the X-co-ordinates of the points, where the graph of
y = p(x) intersects the X-axis.
(i) A linear polynomial has one and only one zero.
(ii) A quadratic polynomial has at most two zeroes.
(iii) A cubic polynomial has at most three zeroes.
(iv) In general, a polynomial of degree n has at most n zeroes.
p
Non-terminating Repeating (or Recurring) Decimals: The decimal expansion obtained from repeats
q
periodically, then it is called non-terminating repeating (or recurring) decimal.
Just divide the numerator by the denominator of a fraction. If you end up with a remainder of 0, you have a
terminating decimal otherwise repeating or recurring decimal.
The sum or difference of a rational and an irrational number is irrational.
The product and quotient of a non-zero rational and an irrational number is irrational.
p
Let x = be a rational number, such that the prime factorization of q is of the form 2m5n, where n and m are non-
q
negative integers. Then, x has a decimal expansion which terminates.
p
Let x be a rational number whose decimal expansion terminates. Then, x can be expressed in the form , where p
q
and q are co-primes and the prime factorization of q is of the form 2m5n, where m and n are non-negative integers.
p
Let x = be a rational number, such that the prime factorization of q is not of the form 2m5n, where n and m are
q
non-negative integers. Then, x has a decimal expansion which is non-terminating repeating.
Know the Formulae
For two positive integers a and b, we have
HCF (a, b) × LCM (a, b) = a × b
a×b
or HCF (a, b) =
LCM ( a , b )
a×b
and LCM (a, b) =
HCF ( a , b )
UNIT II: ALGEBRA
Chapter - 2 : Polynomials
Revision Notes
Zeroes of a Polynomial and Coefficients of
Quadratic Polynomials
Polynomial: An algebraic expression in the form of anxn + an–1xn–1 +.........+ a2x2+a1x + a0, (where n is a whole
number and a0, a1, a2, ........., an are real numbers) is called a polynomial in one variable x of degree n.
Value of a Polynomial at a given point : If p(x) is a polynomial in x and ‘α’ is any real number, then the value
obtained by putting x = α in p(x),
is called the value of p(x) at x = α.
Zero of a Polynomial: A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.
Geometrically, the zeroes of a polynomial p(x) are precisely the X-co-ordinates of the points, where the graph of
y = p(x) intersects the X-axis.
(i) A linear polynomial has one and only one zero.
(ii) A quadratic polynomial has at most two zeroes.
(iii) A cubic polynomial has at most three zeroes.
(iv) In general, a polynomial of degree n has at most n zeroes.
