Key Points - Polynomials

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 +. -. X and 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) 5x^2 - 4x^2 - 6x - 3 is a polynomial in variable x.

(2) 5 + 8x^32 + 4x^-2  is an expression but not a polynomial.
Polynomials are denoted by p(x). q(x) and r(x)etc.
- Coefficients: In the polynomial x 3 + 3x 2 + 3x + 1. coefficient of x 3 . x 2 . x are 1. 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.

- Classification of polynomials on the basis of degree.
Degree  -   Polynomial    - Example
(a) 1  - Linear - x + 1. 2x + 3etc.
(b) 2 - Quadratic - ax 2 + bx + c etc.
(c) 3 - Cubic - x 3 + 3x 2 + 1 etc. etc.
(d) 4 - Biquadratic - x4 −1

Classification of polynomials on the basis of no. of terms

- No. of terms - Polynomial & Examples.
(i) 1     - 1  Monomial - 1/3 
(ii) 2   - Binomial - (3+ 6x), (x - 5y) etc.
(iii) 3   - Trinomial- 2x 2 + 4x + 2 etc. etc.

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(α ) =0. then we say that 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 n ≥ 1 and let a be any real number. When f(x) is divided by (x − a) 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. (i) If f (a) = 0 then (x − a) is factor of f (x) (ii) If (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.