Examples of Constant Polynomials and Their Properties
Understanding a constant polynomial is essential for school algebra exams and building confidence in maths foundations. Students must quickly identify, define, and distinguish these basic polynomials—skills that appear in both board papers and competitive tests. Constant polynomials also set the stage for learning higher-degree algebraic expressions.
Formula Used in Constant Polynomial
The standard formula is: \( f(x) = k \), where k is any real number and there are no variable terms.
Here’s a helpful table to understand constant polynomial more clearly:
Constant Polynomial Table
| Expression | Is It a Constant Polynomial? | Degree |
|---|---|---|
| f(x) = 7 | Yes | 0 |
| f(x) = 0 | Yes (Zero Polynomial) | 0 |
| f(x) = 4x | No | 1 |
| f(x) = -3 | Yes | 0 |
| f(x) = 2x + 5 | No | 1 |
This table shows how the pattern of constant polynomial appears regularly and how to identify one based on definition and degree.
Worked Example – Solving a Problem
1. Let f(x) = -10 be a constant polynomial. Find f(6).Step 1: Identify that f(x) is constant for all values of x.
Step 2: Substitute x = 6 into the formula:
f(6) = -10
Final Answer: f(6) = -10
2. What is the degree of the constant polynomial g(x) = 3.4?
Step 1: Recognize that a constant polynomial has no variable, or it can be written as g(x) = 3.4x0.
Step 2: Highest power of x is 0.
Final Answer: Degree is 0.
3. Is f(x) = 0 a constant polynomial?
Step 1: Yes, this is a constant polynomial called the zero polynomial.
Final Answer: Yes, and its degree is sometimes defined as “not defined” or 0 by convention.
Practice Problems
- Write two examples of constant polynomials and state their degrees.
- Is f(x) = π a constant polynomial? Why?
- Find the value of h(-1) if h(x) = 8.
- Explain why 3x is not a constant polynomial.
Common Mistakes to Avoid
- Confusing constant polynomial with polynomials having variable terms, like 5x.
- Forgetting that the degree of any non-zero constant polynomial is always 0.
- Assuming the zero polynomial’s degree is always zero, when sometimes it is “not defined.”
- Trying to graph a constant polynomial as anything other than a straight, horizontal line.
Real-World Applications
The concept of constant polynomial appears in fields like signal processing (constant outputs), budgeting (fixed costs), and scientific measurements with unchanging results. At Vedantu, students learn how fundamental ideas such as constant polynomials have uses that go beyond classroom maths—and often form the basis of real-life equations and graphs.
We explored the idea of constant polynomial, how to recognize it, apply its properties, and avoid common student mistakes. Practice more on Vedantu to gain confidence in algebraic concepts and score higher in your exams.
For further reading, try:
- Zero Polynomial
- Polynomial Definition
- Difference Between Constants and Variables
- Algebraic Expressions
- Polynomial Equations
FAQs on What Is a Constant Polynomial?
1. What is a constant polynomial?
A constant polynomial is a polynomial that has the same value for all values of the variable. Its general form is f(x) = k, where k is any real number and does not depend on x. This means that the value of the polynomial does not change when you change x.
2. Is 3 a constant polynomial?
Yes, 3 is a constant polynomial because it can be written as f(x) = 3, which represents the same value for any value of x. Here, the degree of the polynomial is 0 because there is no variable term.
3. Why is 7 a constant polynomial?
The number 7 is considered a constant polynomial because it is independent of the variable x. In the form f(x) = 7, the output is always 7 no matter what value is given to x, so it fits the definition of a constant polynomial.
4. What does a constant polynomial look like?
A constant polynomial looks like a single real number, such as f(x) = 5 or f(x) = -2. It has no variable term and remains the same for every value of x. On a graph, it appears as a horizontal straight line.
5. What is the degree of a constant polynomial?
The degree of a constant polynomial is always zero, because there is no variable x raised to any power. For example, in f(x) = 8, the exponent of x is considered to be 0 since it is equal to 8·x0.
6. What is a zero polynomial?
A zero polynomial is a special case of a constant polynomial where all the coefficients are zero, i.e., f(x) = 0 for all values of x. Its degree is sometimes considered undefined or negative infinity.
7. How is a constant polynomial different from a zero polynomial?
A constant polynomial can have any real value except zero, whereas a zero polynomial is always equal to zero for all values of x. The degree of a constant polynomial is 0, but the degree of a zero polynomial is undefined.
8. What are some examples of constant polynomials?
Examples of constant polynomials include f(x) = 4, f(x) = -1, f(x) = 22, and any polynomial of the form f(x) = k, where k is a real number.
9. What does the graph of a constant polynomial look like?
The graph of a constant polynomial is a straight horizontal line parallel to the x-axis at the value k (y = k). For example, the graph of f(x) = 6 is a line parallel to the x-axis and 6 units above it.
10. What is the difference between a constant polynomial and a linear polynomial?
A constant polynomial has degree 0 and always gives the same value, while a linear polynomial has degree 1 and is of the form ax + b, where the value depends on x. For example, f(x) = 7 (constant) versus f(x) = 2x + 3 (linear).
11. What do you mean by a non-zero constant polynomial?
A non-zero constant polynomial is a constant polynomial in which the constant k ≠ 0. For example, f(x) = 2 or f(x) = -5 are non-zero constant polynomials.
12. Constant polynomial meaning in Hindi (कॉन्स्टेंट बहुपद का अर्थ क्या है)?
कॉन्स्टेंट बहुपद (Constant Polynomial) वह बहुपद होता है जिसमें कोई चरों (variable) नहीं होता, अर्थात् उसका मान हमेशा समान रहता है, जैसे f(x) = 5 या f(x) = -3। इसका डिग्री (degree) शून्य (0) होती है।
