1. Introduction
Polynomials can be classified in different ways, mainly based on the number of terms and the degree of the polynomial. Understanding these types helps in identifying, comparing, and solving polynomial expressions easily.
In this topic, we explore the different types of polynomials in a simple, student-friendly manner.
2. Classification Based on Number of Terms
A polynomial is made up of terms. A term is a part of an expression separated by a plus or minus sign. Based on the number of terms, polynomials can be classified into four types:
2.1. Monomial
A monomial is a polynomial with exactly one term.
Examples:
- \(5x\)
- \(-3a^2\)
- \(7\)
2.2. Binomial
A binomial has two terms separated by a plus or minus sign.
Examples:
- \(x + 5\)
- \(3a - 2b\)
- \(4y + y^2\)
2.3. Trinomial
A trinomial has three terms.
Examples:
- \(x^2 + x + 1\)
- \(2m - m^2 + 7\)
- \(3a + 2b - 5\)
2.4. Polynomial (General Case)
A polynomial may have one or more terms. Monomials, binomials, and trinomials are all special types of polynomials.
Examples:
- \(x + 1\)
- \(x^2 - 3x + 2\)
- \(5x^3 + x^2 - x + 4\)
3. Classification Based on Degree
The degree of a polynomial is the highest power of the variable in the polynomial. Based on degree, polynomials are classified into four types:
3.1. Constant Polynomial
A constant polynomial has degree 0.
Examples:
- \(5\)
- \(-12\)
- \(8\)
3.2. Linear Polynomial
A linear polynomial has degree 1.
Examples:
- \(3x + 2\)
- \(y - 4\)
- \(2a + 5\)
3.3. Quadratic Polynomial
A quadratic polynomial has degree 2.
Examples:
- \(x^2 + 3x + 2\)
- \(5y^2 - y - 6\)
- \(2m^2 - 7\)
3.4. Cubic Polynomial
A cubic polynomial has degree 3.
Examples:
- \(x^3 - x + 1\)
- \(3a^3 - 2a^2 + a - 9\)
- \(7p^3\)
4. Example Identification Table
The table below shows different polynomials and their types based on both number of terms and degree.
4.1. Polynomials Table
| Polynomial | Number of Terms | Type by Terms | Degree | Type by Degree |
|---|---|---|---|---|
| \(7x\) | 1 | Monomial | 1 | Linear |
| \(3x + 5\) | 2 | Binomial | 1 | Linear |
| \(x^2 + x + 1\) | 3 | Trinomial | 2 | Quadratic |
| \(4x^3 - x + 2\) | 3 | Trinomial | 3 | Cubic |
5. Common Mistakes
- Confusing number of terms with degree of polynomial.
- Thinking a constant term has no degree (it has degree 0).
- Misidentifying polynomials with more variables.
- Assuming all polynomials must have three terms (only trinomials do).
6. Quick Practice
Identify the type of the following polynomials:
- \(9x^2 - 4x + 1\)
- \(5y + 7\)
- \(-3\)
- \(2a^3 + a - 5\)
- \(x\)
7. Summary
- Polynomials are classified based on number of terms and degree.
- Monomial → 1 term; Binomial → 2 terms; Trinomial → 3 terms.
- Constant → degree 0; Linear → degree 1; Quadratic → degree 2; Cubic → degree 3.
- Identifying types helps in solving polynomial problems more easily.