Polynomials - Revision Notes

 CBSE Class 09 Mathematics

Revision Notes

  1. Polynomials in one Variable
  2. Zeroes of a Polynomial
  3. Remainder Theorem
  4. Factorisation of Polynomials
  5. 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 non-negative integral powers is called a polynomial.

(i) 5x24x26x3 is a polynomial in variable x.

(ii) 5+8x32+4x2 is an expression but not a polynomial.

Polynomials are denoted by p(x), q(x) and r(x) etc.

Coefficients : In the polynomial x3+3x2+3x+1,, coefficient of x3,x2,xare1,3,3 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.




(a) 1



(b) 2



(c) 3



(d) 4



Classification of polynomials on the basis of number of terms

No. of terms

Polynomial & Examples.

(i) 1

Monomial - 5,3x,13yetc.


(ii) 2

Binomial - (3+6x),(x5y) etc.

(iii) 3

Trinomial- 2x2+4x+2etc.


Constant polynomial : A polynomial containing one term only, consisting a constant term is called a constant polynomial.The degree of non-zero 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 p(x) be a polynomial. If p(a)=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 f(x) be a polynomial of degree n1 and let a be any real number. When f(x) is divided by (xa) 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 p(x) is called factor of q(x) divides q(x) 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) (x+y)2(x)2+2xy+(y)2

(ii) (xy)2(x)2-2xy+(y)2

(iii) x2-y2=(x-y)(x+y)

(iv) (x+a)(x+b)= (x)2+(a+b)x+ab


(vi) (x+y)3=x3+y3+3xy(x+y)

(vii) (xy)3=x3-y3-3xy(x-y)

(viii) x3 + y3 + z3 - 3xyz =(x+y+z) (x2 + y2 + z2 - xy - yz - zx)

x3+y3+z3=3xyz if x+y+z=0

(ix) a3+b3=(a+b)(a2ab+b2)