Basic Definitions and Concepts
- 1 Basic Definitions and Concepts
- 1.0.1 Definition
- 1.0.2 Variables
- 1.0.3 Constants
- 1.0.4 Algebraic Expressions
- 1.0.5 Coefficients
- 1.0.6 Degree of a Polynomial
- 1.0.7 Polynomials with various Degrees
- 1.0.8 Constant Polynomial
- 1.0.9 Zero Polynomial
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.
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.
Symbols having fixed numerical values are called constants.
For Example- 2, 3, 1/2, 22/7, etc.
When we combine the variables and constants along with operations +, -, ×, and ÷ (few or all), the result obtained is called an algebraic expression.
Note- This above expression contains 3 terms (Home Work- Name these)
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.
Degree of this polynomial is- 4 (2nd Term i.e., 5x²y² = Sum of powers- 2+2)
Polynomials with various Degrees
- Linear Polynomial – Having degree 1. Ex- x-2
- Quadratic Polynomial- Having degree 2. Ex- y²+x-2
- Cubic Polynomial- Having degree 3. Ex- 8y³+5x²+x
- Biquadratic Polynomial- Having degree 4. Ex- 8y³+5x²y²+x-2
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.
A polynomial containing only one term, i.e, zero is called a zero polynomial. Its degree is not defined.