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1. Mathematics
Class: IX

2. Algebraic Expression: In mathematics, an algebraic expression is
an expression built up from integer constants, variables, and
the algebraic operations (addition, subtraction, multiplication,
division and exponentiation by an exponent that is a rational number).
e.g.- (a) 2x3
– 4x2
+ 6x – 3 is a polynomial in one variable x.
(b) 8p7
+ 4p5
+ 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.
e.g.: 3x2
+ 4y
Constants: A symbol having a fixed numerical value is called a
constant.
e.g.: In polynomial 3x2
+ 4y, 3 and 4 are the constants.
Variables: A symbol which may be assigned different
numerical values is called as a variables.
e.g.: In polynomial 3x2
+ 4y, x and y are the variables.

4. Degree: The highest power of a variable in the polynomial is
called degree of that polynomial.
e.g.: 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 non-zero constant polynomial is zero.
e.g.: 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. Like Terms: two or more having same type of variable and same
power on them are said to be like terms for example 3x , -7/2x,
8/9x , are like terms.
Unlike Terms: terms if they are not like then they are known as
unlike terms for example 7a , 8b , 19/3c are unlike terms.

6. Monomial: Algebraic expression that consists only one term is
called monomial. e.g.: 3x2
, 4y
Binomial: Algebraic expression that consists two terms is called
binomial. e.g.: 3x2
- 8y2
, x2
- 4
Trinomial: Algebraic expression that consists three terms is
called trinomial. e.g.: z2
+ 4z - 9
Polynomial: Algebraic expression that consists many terms is
called polynomial. e.g.: 3x3
+ 4x2
- 2x - 7

7. Types of polynomial on the basis of degree are :
Linear polynomial: A polynomial of degree 1 is called a linear
polynomial. e.g.: 3x, 4y
Quadratic polynomial: A polynomial of degree 2 is called a
quadratic polynomial. e.g.: x2
+ 5, y2
- 9
Cubic polynomial : A polynomial of degree 3 is called a cubic
polynomial. e.g.: x3
- x2
+ 5x - 7
Bi-quadratic polynomial : A polynomial of degree 4 is called
a bi-quadratic polynomial. e.g.: 3x4
- x3
+ x2
+ 5x - 7

8. 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

9. How to convert a polynomial into standard
form?
OR

10. 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 g(x) and the remainder is r.
Therefore, f(x) = (x-a) x g(x) + r (i)
Putting x=a in equation (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).

11. Let f(x) be a polynomial of degree n>1 and let a be
any real number. If f(a) = 0 then (x-a) is a factor of f(x).
Proof: Let f(a) = 0
On dividing f(x) by (x-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) x g(x) + f(a)
f(x) = (x-a) x g(x) as f(a)=0 (given)
(x-a) is a factor of f(x).

12.  (a + b)2
= a2
+ 2ab + b2
 (a – b)2
= a2
– 2ab + b2
 a2
– b2
= (a - b) (a – b)
 (x + a) (x + b) = x2
+ (a + b)x + ab
 (a + b + c)2
= a2
+ b2
+ c2
+ 2ab + 2bc + 2ca
 (a + b)3
= a3
+ b3
+ 3ab (a + b)
 (a – b)3
= a3
– b3
– 3ab (a – b)
 a3
+ b3
= (a + b) (a2
– ab + b2
)
 a3
– b3
= (a – b) (a2
+ ab + b2
)
 a3
+ b3
+ c3
– 3abc = (a + b + c) (a2
+ b2
+ c2
– ab – bc – ca)
 a3
+ b3
+ c3
= 3abc , If a + b + c = 0

13. THANKS