How to Find the Degree of a Polynomial (With Examples)
The concept of degree of polynomial plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding the degree helps students classify, solve, and analyze polynomials easily, especially for competitive exams and foundational algebra topics.
What Is Degree of Polynomial?
A degree of polynomial is defined as the highest power (exponent) of the variable in a polynomial expression with a non-zero coefficient. For example, in the polynomial \( f(x) = 4x^3 + 2x + 7 \), the highest exponent is 3, so its degree is 3. You’ll find this concept applied in solving equations, analyzing polynomial graphs, and even real-life modeling of scenarios needing algebra.
Key Formula for Degree of Polynomial
Here’s the standard formula: Degree = Highest exponent of the variable among all the terms with non-zero coefficients in the polynomial
Degree Name Table and Classification
| Degree | Polynomial Name | General Form | Example |
|---|---|---|---|
| 0 | Constant | \( f(x) = c \) | 7 |
| 1 | Linear | \( f(x) = ax + b \) | 2x + 5 |
| 2 | Quadratic | \( f(x) = ax^2 + bx + c \) | 3x2 + x + 1 |
| 3 | Cubic | \( f(x) = ax^3 + bx^2 + cx + d \) | x3 - 4x + 6 |
| 4 | Bi-quadratic/Quartic | \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \) | 2x4+ 5x3-1 |
Special Cases: Constant and Zero Polynomial
Constant Polynomial: Any polynomial like \( f(x) = 8 \), where there is no variable, has degree 0. This is because it can be written as \( 8x^0 \).
Zero Polynomial: The polynomial where all coefficients are zero, i.e., \( f(x) = 0 \), is called the zero polynomial. Its degree is undefined as there’s no non-zero term.
How to Find Degree of a Polynomial: Step-by-Step Illustration
- Write out the polynomial, e.g., \( 3x^4 + 2x^2 + x + 7 \).
Focus on each term: \( 3x^4, 2x^2, x, 7 \).
- Look at the exponents: 4 (from \( x^4 \)), 2, 1 (from \( x \)), and 0 (constant).
- The highest exponent is the degree: Here, 4.
Degree in Multivariable Polynomials
For polynomials with more than one variable, the degree is the highest sum of exponents in any term. For example, \( 5x^2y^3 + 4xy \) has: (2+3)=5 and (1+1)=2. So, the degree is 5.
Common Mistakes to Avoid
- Forgetting to group like terms first before picking the highest power.
- Thinking that the degree is the sum of all exponents—only the largest, or sum for multivariable terms.
- Assigning a degree to the zero polynomial (it is always undefined).
Relation to Other Polynomial Concepts
The degree of polynomial connects closely with polynomial basics, types of Polynomials, and understanding quadratic polynomials (degree 2). Mastering degrees helps with graphing functions, predicting number of roots, and polynomial division.
Step-by-Step Example Problems
Example 1: Find the degree of \( f(x) = 5x^6 + 2x^3 - x + 4 \)
1. List the exponents for each term: 6, 3, 1, 02. The highest exponent is 6
3. Degree = 6
Example 2: What is the degree of \( 4xy^2 + y^3 + 5x \)?
1. Get sum of exponents for each term:2. Highest sum is 3
3. Degree = 3
Key Degree Rules and Summary Table
| Operation | Formula | Example |
|---|---|---|
| Addition | deg(P + Q) ≤ max[deg(P), deg(Q)] | deg(\( x^4 + x^2 \)) = 4 |
| Multiplication | deg(P × Q) = deg(P) + deg(Q) | deg(\( x^3 \) × \( x^2 \)) = 5 |
| Constant | deg(k) = 0 (k ≠ 0) | deg(9) = 0 |
| Zero Polynomial | Undefined | deg(0) = undefined |
Try These Yourself
- Find the degree of \( 9x^2y^4 + 5x^3 + 10y \).
- Is the degree of \( 0 \) defined or not?
- What is the degree of the constant polynomial \( 12 \)?
- For \( f(x, y) = x^2y^5 - x^4 \), what is the degree?
Classroom Tip
A quick way to remember: “Look for the biggest exponent in any term — that’s the degree.” For multivariable expressions, add exponents in a term for the biggest sum. Vedantu’s live classes explain these tips using fun visuals and quizzes!
We explored degree of polynomial—from definition, formula, examples, errors, and tips. With practice, classifying polynomials and solving equation questions becomes much faster. For complete learning journeys and even more solved MCQs, keep practicing with Vedantu’s interactive resources.
Useful links: What is a Polynomial?, Quadratic Polynomial, Zero Polynomial Explained
FAQs on Degree of Polynomial in Maths: Meaning, How to Find & Tips
1. What is the degree of the polynomial?
The degree of a polynomial is the highest power (exponent) of the variable in the polynomial expression. For a single-variable polynomial, it is determined by looking at the largest exponent present among all its terms. For example, in $3x^{4} + 2x^{2} - 7$, the term $3x^{4}$ has the highest exponent (4), so the degree is 4. Understanding the degree is important as it helps determine the number of solutions and the general shape of the polynomial’s graph. At Vedantu, our expert teachers explain polynomial degrees in detail through live interactive classes and step-by-step solved examples.
2. What is the degree of polynomial 5x 3 4x 2 7x?
The polynomial $5x^3 + 4x^2 + 7x$ contains three terms: $5x^3$, $4x^2$, and $7x$. To find the degree, identify the term with the highest exponent:
- $5x^3$ has degree 3
- $4x^2$ has degree 2
- $7x$ has degree 1
3. What is the degree of the polynomial 2x 4 3?
For the polynomial $2x^4 + 3$, there are two terms: $2x^4$ and $3$. The exponent of $x$ in $2x^4$ is 4, and $3$ is a constant term (degree 0). Therefore, the highest exponent is 4, making the degree of this polynomial 4. Our Vedantu programs offer easy-to-understand lessons on polynomials, including tips for quickly determining polynomial degrees.
4. What is the degree of x5 x4 3?
Given the polynomial $x^5 + x^4 + 3$, we look for the term with the highest exponent. Here, $x^5$ has an exponent of 5, $x^4$ has 4, and $3$ has 0. Thus, the degree of the polynomial is 5. At Vedantu, students can access numerous examples and practice problems to strengthen their command on polynomial degrees.
5. How do you find the degree of a polynomial with multiple variables?
To find the degree of a polynomial with multiple variables, first determine the degree of each term by adding the exponents of all variables in that term, then choose the highest sum among all terms. For example, in $3x^2y^3 + 4xy^2 + 5$, the degrees of the terms are:
- $3x^2y^3$: $2 + 3 = 5$
- $4xy^2$: $1 + 2 = 3$
- $5$: $0$
6. What is the relationship between the degree of a polynomial and its number of roots?
The degree of a polynomial indicates the maximum number of roots (solutions or zeros) the polynomial can have. A polynomial of degree $n$ can have up to $n$ real or complex roots in total. For instance, a cubic polynomial (degree 3) can have at most 3 roots. At Vedantu, skilled teachers help students understand the connection between polynomial degree, number of roots, and how this impacts graphing and problem-solving.
7. Can a polynomial have a negative degree?
No, a polynomial cannot have a negative degree. The degree of a polynomial is defined as a non-negative integer—the highest exponent of the variable(s) in the expression. Negative or fractional exponents result in an expression that is not considered a polynomial. At Vedantu, our curriculum clearly explains polynomial definitions and provides plenty of examples to illustrate this rule.
8. Why is the degree of a constant polynomial considered zero?
A constant polynomial (such as $5$, $-7$, or any fixed number) does not contain any variable. By definition, the degree of such a polynomial is zero because it can be written as $5x^0$, where the exponent is 0. This key concept is regularly detailed in Vedantu’s foundational math classes to help students develop a strong base in algebra.
9. How does the degree of a polynomial affect its graph?
The degree of a polynomial has a direct impact on the shape and behavior of its graph:
- The degree determines the maximum number of turning points (which is at most $n-1$ for a degree $n$ polynomial).
- The end behavior (how the graph grows as $x$ approaches infinity or negative infinity) is also determined by the degree and the leading coefficient.
