The algebraic expression involving variables, coefficients, and constants is called a polynomials. The variables, constants, or products of variables and constants form the different terms of the algebraic expression. These terms are connected through arithmetic operations like addition, subtraction, and multiplication to form a polynomial expression.
Types of Polynomials
Polynomials can be of different types based on the number of terms and degree of variables. These are explained as follows:
- Monomials: Polynomials containing only a single term are called monomial.
For example, 3x, -8y2
- Binomials: Polynomials containing two terms are called binomials.
For example, x + 2, y2– 5
- Trinomials: Polynomials containing three terms are called trinomials.
For example, 4x + 3y- 8, 2y2 – 9x + 23
Different Parts of a Polynomial
The name polynomial has two parts, one is ‘poly’ which means ‘many’, and the other is ‘nominal’ which means ‘terms’. So a polynomial is a combination of many terms but the number of terms is fixed. A polynomial consists of various elements which are:
- Variable: Any unknown quantity that can change value according to the conditions and boundary set in the expression.
- Constant: A quantity that doesn’t change the value under the given conditions.
- Coefficients: The fixed number (positive or negative) that is multiplied with variables.
- Exponents: The degree of the variable which means the number of times to which a variable is to be multiplied by itself.
Example
The example of a polynomial is 2x2– 5y + 16
The polynomial has three terms involving two variables and one constant.
Here, x and y are variables, the coefficient of x is 2, and the coefficient of y is -5. The exponent of x is 2 and the exponent of y is 1. The constant term is 16.
Like and Unlike Terms
In a polynomial, the terms which have the same variable and same exponent are called like terms. Terms that have different variables and different exponents are called, unlike terms. When two or more polynomials are added or subtracted, it is necessary to combine the like terms and then perform addition or subtraction with their coefficients.
Important Points About Polynomial
- The polynomial contains at least one variable or constant.
- The polynomial doesn’t contain any ‘=’ sign.
- The polynomial only involves addition, subtraction, and multiplication operation.
- The polynomial doesn’t contain any term which denotes division by variable.
- In a polynomial expression, the exponents of the variable should be a positive whole number.
- The addition and subtraction of polynomials are done with like terms only.
Degree of Polynomial
In a polynomial, the highest exponent value of the variable determines the degree of the polynomial. In case the polynomial contains more than one variable, we have to find the degree of each term involving variables and the highest value gives the degree of that polynomial.
Classification of Polynomials
Polynomials can be classified according to the degree of polynomial. These are as follows:
- Zero Polynomial: The degree of a polynomial is zero means the coefficients of the variables are zero which means it contains only a constant.
- Linear Polynomial: Any polynomial with degree 1 is a linear polynomial. For example, 3x + y-10 is a linear polynomial.
- Quadratic Polynomial: Any polynomial with degree 2 is called a quadratic polynomial.
For example, 2y2 +15 is a quadratic polynomial. - Cubic Polynomial: Any polynomial with degree 3 is called a cubic polynomial.
For example, 7x3– 56 is a cubic polynomial.
Example
Let’s consider a polynomial 3x3+ 6y + 8. It has a degree of 3.
Again, we can find the degree of the polynomial 5x3y2 – 2xy3 by finding the highest exponent from each term. In this case, the degree of the polynomial is 5.
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