Basic Definitions and Concepts

Definition

A polynomial is an algebraic expression in which the variables involving are having non-negative integral powers. 

Examples- 2x³+5x+2, 8y³+5xy+x-2 etc.

Ok, lets now define the term Algebraic Expressions, variables and constants. 

Variables

A symbol (x,y,z,a,b,g,h etc.) that can be assigned different numerical values.

Example- Area of a triangle = ½×base(b)×height(h), Here ‘b’ and ‘h’ can have different numerical values. So, ‘b’ and ‘h’ are variables. 

Constants

 Symbols having fixed numerical values are called constants. 

For Example- 2, 3, 1/2, 22/7, etc.

Algebraic Expressions

When we combine the variables and constants along with operations +, -, ×, and ÷  (few or all), the result obtained is called an algebraic expression.

Example- 2x³+5x+2, 

Note- This above expression contains 3 terms (Home Work- Name these)

Coefficients

The numerical values associated with the variables in a polynomial are called coefficients. For example- In the polynomial,  2x³+5x²+x+2 the coefficients of x³, x², x are 2, 5 and 1 respectively whereas 2 is the constant term.

Degree of a Polynomial

  • Degree of a Polynomial of one Variable

 If a polynomial is in one variable, then the variable with highest non-negative integral power is called its degree.

Ex- x³+8x²+x+2. Here there are 4 terms namely x³, 8x², x, 2 and the variable with the highest power is x³. So the degree of the above polynomial is 3.

  • Degree of a Polynomial of two or more Variables

 In this case, we add the powers of the variables in each term, and the highest sum obtained is called its degree.

Ex- 8y³+5x²y²+x-2.

Degree of this polynomial is- 4 (2nd Term i.e., 5x²y² = Sum of powers- 2+2)

Polynomials with various Degrees

  1. Linear Polynomial – Having degree 1. Ex- x-2
  2. Quadratic Polynomial- Having degree 2. Ex- y²+x-2
  3. Cubic Polynomial- Having degree 3. Ex- 8y³+5x²+x
  4. Biquadratic Polynomial- Having degree 4. Ex- 8y³+5x²y²+x-2

Constant Polynomial

A polynomial containing only one term, i.e, a constant term is called a constant polynomial. Its degree is zero.

Ex- 2, 1/8 etc. 

Zero Polynomial

A polynomial containing only one term, i.e, zero is called a zero polynomial. Its degree is not defined.

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