1. Introduction
A polynomial is an algebraic expression consisting of terms formed by variables and constants combined using addition, subtraction, and multiplication. The degree of a polynomial is an important concept because it tells us the highest power of the variable in the expression.
Understanding the degree helps in classifying polynomials, comparing them, and solving problems involving polynomial equations.
2. What is the Degree of a Polynomial?
The degree of a polynomial is the highest power of the variable in the polynomial. If there are multiple variables, the degree of a term is the sum of the powers of all variables in that term, and the degree of the polynomial is the maximum of these.
Examples:
- The degree of \(5x^3\) is 3.
- The degree of \(4x^2 + 7x - 9\) is 2.
- The degree of \(3xy + 2x\) is 2 (since \(xy = x^1y^1\), degree = 1 + 1 = 2).
2.1. Degree of a Monomial
A monomial is a single-term expression. Its degree is simply the sum of powers of the variable(s) in that term.
Examples:
- \(7x^4\): degree = 4
- \(3x^2y\): degree = 2 + 1 = 3
- \(-5ab^2c\): degree = 1 + 2 + 1 = 4
2.2. Degree of a Polynomial
To find the degree of a polynomial, find the degree of each term and then take the largest one.
Example:
Polynomial: \(4x^3 + 7x^2 - 9\)
- Degree of \(4x^3\): 3
- Degree of \(7x^2\): 2
- Degree of constant \(-9\): 0
Degree of polynomial = 3
2.2.1. Degree Comparison Table
| Polynomial | Degrees of Terms | Degree |
|---|---|---|
| \(5x + 2\) | 1, 0 | 1 |
| \(2x^2 + 7x - 3\) | 2, 1, 0 | 2 |
| \(3xy + 2x^3\) | 2, 3 | 3 |
3. Degree in Polynomials with More Than One Variable
In multivariable polynomials, the degree of each term is the sum of exponents of all variables in that term.
Example:
Polynomial: \(4x^2y + 3xy^3 + 5\)
- Term \(4x^2y\): degree = 2 + 1 = 3
- Term \(3xy^3\): degree = 1 + 3 = 4
- Term \(5\): degree = 0
Degree of polynomial = 4
3.1. More Examples
- \(6a^2b^2\): degree = 2 + 2 = 4
- \(2pqr\): degree = 1 + 1 + 1 = 3
- \(7x^3y^2z\): degree = 3 + 2 + 1 = 6
4. Special Cases
Certain polynomials require special attention while finding their degree.
4.1. Zero Polynomial
A zero polynomial is simply 0 (all coefficients are zero). Its degree is not defined because it has no non-zero term.
4.2. Constant Polynomial
A constant polynomial is of the form \(c\) where \(c \neq 0\). Its degree is 0.
Examples:
- \(5\): degree = 0
- \(-12\): degree = 0
5. Common Mistakes
- Ignoring variables in multivariable expressions.
- Thinking constant term has no degree (it has degree 0).
- Adding degrees of different terms (you should pick the highest, not add).
- Forgetting that degree of 0 polynomial is undefined.
6. Quick Practice
Find the degree of the following polynomials:
- \(7x^5 - 3x^2 + 1\)
- \(4y^3 + 6y - 8\)
- \(2xy + 5x\)
- \(3a^2b + 2ab^2\)
- \(0\) (zero polynomial)
7. Summary
- Degree of a polynomial is the highest power of the variable in any term.
- Degree of a monomial is the sum of exponents of all variables in the term.
- For multivariable polynomials, calculate degree of each term and choose the largest.
- Constant polynomials have degree 0; zero polynomial has no defined degree.