Made By : Sejal Agarwal
School : Ryan International School
Class : 9th
Subject : Maths
Submitted To : Ms Madhu Kanta
Algebraic Expression : The combination of constants and
variables are called algebraic expressions.
E.g.- (a) 2x3
+6x–3 is a polynomial in one variable x.
(b) 8p7+4p2+11p3-9p is a polynomial in one variable p.
+9x5 is an expression but not a polynomial
since it contains a term x4/5
, where 4/5
a non-negative integer.
Polynomials : An algebraic expression in which the
variable involved have only non –negative integral
powers is called a polynomial.
Eg. : 3x + 4y
Constants : A symbol having a fixed numerical value is
called a constant.
Eg. : In polynomial 3x + 4y ,3 and 4 are the constants.
Variables : A symbol which may be assigned different
numerical values is called as a variables.
Eg. : In polynomial 3x + 4y , x and y are the variables.
Degree : The highest power of a variable in the
polynomial is called degree of that polynomial.
Eg. : 5x2 + 3 , here the degree is 2.
Constant polynomial : A polynomial containing one
term only , consisting of a constant is called a
The degree of a nonzero constant polynomial is zero.
Eg. : 3 , -5 , 7/8 , etc. , are all constant polynomials.
Zero polynomial : A polynomial consisting one term
only , namely zero only , is called a zero polynomial.
The degree of a zero polynomial is not defined.
Monomial : Algebric expression that consists only one
term is called monomial.
Binomial : Algebric expression that consists two terms
is called binomial.
Trinomial : Algebric expression that consists three
terms is called trinomial.
Polynomial : Algebric expression that consists many
terms is called polynomial.
Types of polynomial on the basis of degree are :
Linear polynomial: A polynomial of degree 1 is called
a linear polynomial.
Quadratic polynomial: A polynomial of degree 2 is
called a quadratic polynomial.
Cubic polynomial : A polynomial of degree 3 is called
a cubic polynomial.
Biquadratic polynomial : A polynomial of degree 4 is
called a biquadratic polynomial.
Polynomials Degree Classify by
Classify by no.
5 0 Constant Monomial
2x - 4 1 Linear Binomial
+ x 2 Quadratic Binomial
+ 1 3 Cubic Trinomial
Phase 1Phase 1 Phase 2Phase 2
To rewrite a
terms of the
starting with the
term and ending
with the lowest
The leading coefficient,
the coefficient of the
first term in a
polynomial written in
standard form, should
How to convert a polynomial into standard form?
Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1]
degree r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).
Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by
remainder theorem, when f(x) is divided by (x-a), then
the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore f(a)=0(given]
(x-a) is a factor of f(x).
Some common identities used to factorize polynomials
A real number ‘a’ is a zero of a polynomial p(x) if
p(a)=0. In this case, a is also called a root of the
Every linear polynomial in one variable has a unique
zero, a non-zero constant polynomial has no zero, and
every real number is a zero of the zero polynomial.