Polynomials  Revision Notes
CBSE Class 09 Mathematics
 Polynomials in one Variable
 Zeroes of a Polynomial
 Remainder Theorem
 Factorisation of Polynomials
 Algebraic Identities
Constants : A symbol having a fixed numerical value is called a constant.
Variables : A symbol which may be assigned different numerical values is known as variable.
Algebraic expressions : A combination of constants and variables connected by some or all of the operations +, , *,/ is known as algebraic expression.
Terms : The several parts of an algebraic expression separated by '+' or '' operations are called the terms of the expression.
Polynomials : An algebraic expression in which the variables involved have only nonnegative integral powers is called a polynomial.
(i) is a polynomial in variable x.
(ii) is an expression but not a polynomial.
Polynomials are denoted by p(x), q(x) and r(x) etc.
Coefficients : In the polynomial , coefficient of respectively and we also say that +1 is the constant term in it.
Degree of a polynomial in one variable: In case of a polynomial in one variable the highest power of the variable is called the degree of the polynomial.
A polynomial of degree n has n roots.
Classification of polynomials on the basis of degree.
degree  Polynomial  Example 
(a) 1  Linear  
(b) 2  Quadratic  
(c) 3  Cubic  
(d) 4  Biquadratic 
Classification of polynomials on the basis of number of terms
No. of terms  Polynomial & Examples. 
(i) 1  Monomial 

(ii) 2  Binomial  etc. 
(iii) 3  Trinomial

Constant polynomial : A polynomial containing one term only, consisting a constant term is called a constant polynomial.The degree of nonzero constant polynomial is zero.
Zero polynomial : A polynomial consisting of one term, namely zero only is called a zero polynomial.
The degree of zero polynomial is not defined.
Zeroes of a polynomial : Let be a polynomial. If =0, then we say that "a" is a zero of the polynomial of p(x).
Remark : Finding the zeroes of polynomial p(x) means solving the equation p(x)=0.
Remainder theorem : Let be a polynomial of degree and let a be any real number. When f(x) is divided by then the remainder is f ( a)
Factor theorem : 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 factor of f(x)
If f(x – a) is factor of f(x) then f(a) = 0
Factor : A polynomial is called factor of divides exactly.
Factorization : To express a given polynomial as the product of polynomials each of
degree less than that of the given polynomial such that no such a factor has a factor of
lower degree, is called factorization.
Some algebraic identities useful in factorization:
(i) = +2xy+
(ii) = 2xy+
(iii) =(xy)(x+y)
(iv) (x+a)(x+b)= +(a+b)x+ab
(v)=+2xy+2yz+2zx
(vi) =++3xy(x+y)
(vii) =3xy(xy)
(viii) + +  3xyz =(x+y+z) ( + +  xy  yz  zx)
++=3xyz if x+y+z=0
(ix)
(x)