How To Find The Degree Of A Polynomial With Definition And 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 Algebra
1. What is the degree of a polynomial?
The degree of a polynomial is the highest power of the variable with a non-zero coefficient in the polynomial.
- In 5x³ + 2x² − x + 7, the highest exponent is 3, so the degree is 3.
- The term with the highest power is called the leading term.
- The exponent must be a whole number (0, 1, 2, 3, ...).
2. How do you find the degree of a polynomial?
To find the degree of a polynomial, identify the term with the highest exponent of the variable.
- Step 1: Write the polynomial in standard form (descending powers).
- Step 2: Look for the highest exponent with a non-zero coefficient.
- Step 3: That exponent is the degree.
3. What is the degree of a constant polynomial?
The degree of a non-zero constant polynomial is 0.
- Example: In 7, there is no variable, which can be written as 7x⁰.
- Since the exponent is 0, the degree is 0.
- The constant polynomial 0 has no defined degree (sometimes called undefined).
4. What is the degree of the zero polynomial?
The degree of the zero polynomial is undefined.
- The zero polynomial is written as 0.
- It has no non-zero term, so there is no highest exponent.
- For this reason, its degree is not defined.
5. What is the degree of a polynomial with two variables?
The degree of a polynomial in two variables is the highest sum of exponents in any term.
- Example: In 3x²y + 4xy³ + y², calculate total powers:
- x²y → 2 + 1 = 3
- xy³ → 1 + 3 = 4
- y² → 2
6. What is the formula for the degree of a polynomial?
The degree of a polynomial is given by the highest exponent of the variable (or highest sum of exponents in multivariable terms).
- For one variable: Degree = max exponent of x.
- For multiple variables: Degree = max (sum of exponents in a term).
7. What is the difference between degree and order of a polynomial?
The degree of a polynomial is the highest power of the variable, while the order refers to how the terms are arranged.
- Degree depends on the highest exponent.
- Order can be ascending (lowest to highest power).
- Or descending (highest to lowest power), called standard form.
8. Can you give an example of a quadratic and cubic polynomial based on degree?
A polynomial is classified by degree: quadratic has degree 2 and cubic has degree 3.
- Quadratic polynomial: 2x² + 3x + 1 (degree = 2)
- Cubic polynomial: x³ − 4x + 5 (degree = 3)
9. How does the degree of a polynomial affect its graph?
The degree of a polynomial determines the maximum number of turning points and end behavior of its graph.
- A degree n polynomial has at most n − 1 turning points.
- If the degree is even, both ends move in the same direction.
- If the degree is odd, the ends move in opposite directions.
10. What are common mistakes when finding the degree of a polynomial?
Common mistakes when finding the degree of a polynomial include ignoring zero coefficients and miscalculating exponents.
- Forgetting to write the polynomial in standard form.
- Choosing the largest coefficient instead of the highest exponent.
- Not adding exponents correctly in multivariable terms.
- Assuming the zero polynomial has degree 0 (it is undefined).
