The document discusses key concepts related to polynomials including: 1) Algebraic expressions, polynomials, monomials, binomials, trinomials, and their definitions. 2) Constants, variables, and the degree of a polynomial. 3) How to classify polynomials based on degree and number of terms. 4) How to rewrite a polynomial in standard form by arranging terms by descending degree. 5) Remainder and factor theorems related to polynomials. 6) Common identities used to factorize polynomials.

1. Made By : Sejal Agarwal
School : Ryan International School
Class : 9th
Subject : Maths
Submitted To : Ms Madhu Kanta

2. Algebraic Expression : The combination of constants and
variables are called algebraic expressions.
E.g.- (a) 2x3
–4x2
+6x–3 is a polynomial in one variable x.
(b) 8p7+4p2+11p3-9p is a polynomial in one variable p.
(c) 4+7x4/5
+9x5 is an expression but not a polynomial
since it contains a term x4/5
, where 4/5
is not
a non-negative integer.

3. 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.

4. 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
constant polynomial.
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.

5. 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.

6. 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.

7. Polynomials Degree Classify by
degree
Classify by no.
of terms.
5 0 Constant Monomial
2x - 4 1 Linear Binomial
3x2
+ x 2 Quadratic Binomial
x3
- 4x2
+ 1 3 Cubic Trinomial

8. Phase 1Phase 1 Phase 2Phase 2
To rewrite a
polynomial in
standard form,
rearrange the
terms of the
polynomial
starting with the
largest degree
term and ending
with the lowest
degree term.
The leading coefficient,
the coefficient of the
first term in a
polynomial written in
standard form, should
be positive.
How to convert a polynomial into standard form?

9. 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
is r(x).
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).

10. 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).

11. Some common identities used to factorize polynomials
(x+a)(x+b)=x2+(a+b)x+ab(a+b)2
=a2
+b2
+2ab (a-b)2
=a2
+b2
-2ab a2
-b2
=(a+b)(a-b)

12. Advanced identities used to factorize polynomials
(x+y+z)2
=x2
+y2
+z2
+2xy+2yz+2zx
(x-y)3
=x3
-y3
-
3xy(x-y)
(x+y)3
=x3
+y3
+
3xy(x+y)
x3
+y3
=(x+y) *
(x2
+y2
-xy) x3
-y3
=(x+y) *
(x2
+y2
+xy)

13.  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
equation p(x)=0.
 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.