100 LearnersLast updated on August 27th, 2025
Constant Polynomial
A constant polynomial is an algebraic expression that is made up of only one fixed number and no variables. It is written in the form f(x) = k, where k is a real number.
What is a Constant Polynomial?
A constant polynomial is an algebraic expression that contains only a constant term. The degree of a constant polynomial is zero if the constant is non-zero. For the zero polynomial (f(x) = 0), the degree is usually considered undefined, though some conventions assign it a degree of 0. A constant polynomial is written as f(x) = k, where k is a real number. For example, the constant polynomial f(x) = 7 is represented on a graph as a horizontal line at y = 7.
What is the Degree of a Constant Polynomial?
The constant polynomial is one where the highest power of the variable is zero. A constant polynomial has no variable term, which means the highest power of the variable is 0. The degree is the highest exponent of x with a non-zero coefficient; the degree of a constant polynomial is 0. The degree of a constant polynomial is zero if the constant is non-zero. If the constant is zero, its degree is undefined and it is called a zero polynomial.
How to Represent Constant Polynomial in Graph?
As seen in the previous section a constant polynomial has the form f(x) = k, where k is a real number (e.g., 2, 4, -6, 0.8) and no variable terms are present. In a constant polynomial, the graph appears as a horizontal line parallel to the x-axis, intersecting the y-axis at y = k.
The graph above shows the constant polynomial f(x) = 6. No matter what the value of x is, the corresponding output is always 6.
Difference between Constant Polynomial and Zero Polynomial
Let’s compare the constant polynomials and zero polynomials of their properties, and see how they are different.
|
Features |
Constant polynomial |
Zero polynomial |
|
Definition |
A constant polynomial has a fixed non-zero value and no variable. |
A zero polynomial is a polynomial in which all coefficients are zero |
|
Standard form |
f(x) = k, where k is a real number |
f(x) = 0 |
|
Degree |
Zero degree |
The degree is undefined. |
|
Graph shape |
Its graph is a horizontal line parallel to the x-axis. |
Its graph is the x-axis itself. |
Real-Life Applications of Constant Polynomial
Polynomials play the main role in day-to-day life. Polynomials are used in various applications, like designing a bridge, computer graphics, and more. Here are some applications given below.
Designing Structures: Engineers use a polynomial to model how the bridge reacts to loads and strains. For example, engineers use polynomial equations to calculate how much a bridge beam bends under the weight of cars and trucks. This helps to handle the weight that the bridge is supposed to carry.
Computer graphics: In computer graphics, the polynomials are used to create 3D objects and shapes. For example, a polynomial equation can help to describe how the surface of a car looks in a 3D movie or game, making it appear realistic and smooth.
Finance and Economics: Polynomials are used by financial analysts to model the market patterns. For example, the polynomials can be used to check how a stock's price has changed over time.
Projectile Motion: A polynomial equation can model the path of a thrown ball, incorporating both its initial velocity and the constant downward pull of gravity.
Image Manipulation: Polynomials are used in digital image processing as they make the image bigger or smaller uniformly (keeping proportions the same) or non-uniformly (changing proportions).
Common Mistakes and How to Avoid Them on Constant Polynomial
Some students make mistakes without realizing it. Here are some common mistakes and tips to avoid them. Understanding these mistakes helps build a strong foundation in constant polynomials.
Mistake 1
Confusing a constant polynomial with one that has a variable
Students mistakenly think that an expression, like f(x) = 5x, is a constant polynomial. This is wrong because the constant polynomial contains the variable x. A constant polynomial has no variable at all. It looks like f(x) = 7, which always has the same value.
Mistake 2
Confusing the degree of the constant polynomial
Some students mistakenly think that a constant polynomial like f(x) = 9 has a degree other than zero. The degree of a polynomial is the highest exponent of the variable in the expression. In f(x) = 9. So, it has an exponent, it is just zero.
Mistake 3
Expecting constant polynomials to change with x
Students sometimes try to substitute different values of x into a constant polynomial like f(x) = k, expecting different answers. For example, f(x) = 4 always gives the same output, no matter what x is. Understanding that the graph of such a function is a horizontal line, so every x-value gives the same y-value.
Mistake 4
Miswriting the standard form
Students may write the standard form mistakenly when they are trying to solve the problem quickly, like writing f = x = k, instead of f(x) = k.
Mistake 5
Thinking polynomials must always have variables
Students may think that polynomials must always have variables. While most polynomials do have variables, a constant polynomial does not contain variables.
Solved Examples on Constant Polynomial
Problem 1
Find the degree of the polynomial f(x) = 7.
Degree = 0
Explanation
A constant polynomial has no variable term, and the value does not change.
The degree of a constant polynomial is 0.
Problem 2
Evaluate f(x) = −3 at x = 5.
f(5) = −3
Explanation
The f(x) = −3 is a constant, the value of the function does not change with the value of x.
So, f(5) = −3
Problem 3
Sketch the graph of f(x) = 4.
A horizontal line at y = 4
Explanation
This is a horizontal line where the y-value is always 4.
No matter what x value you choose, f(x) = 4.
Problem 4
Add the constant polynomials f (x) = 5 and g(x) = -2
f(x) + g(x) = 3
Explanation
f(x) + g(x)= 5 + (−2) = 3
The answer is f(x) + g(x) = 3.
Problem 5
Multiply the constant polynomials f(x) = 6 and g(x) = −4.
f(x) × g(x) =−24
Explanation
f(x) × g(x)= 6 × (−4) = −24
FAQs on Constant Polynomial
1.What is a constant polynomial?
2.What is the degree of a constant polynomial?
3.Can a constant polynomial be negative?
4.Why is 7 considered a constant polynomial?
5.Is zero polynomial a constant polynomial?
6.How does learning Algebra help students in Vietnam make better decisions in daily life?
7.How can cultural or local activities in Vietnam support learning Algebra topics such as Constant Polynomial ?
8.How do technology and digital tools in Vietnam support learning Algebra and Constant Polynomial ?
9.Does learning Algebra support future career opportunities for students in Vietnam?
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