Constant Polynomial

What Is a Constant Polynomial?

A constant polynomial is a polynomial of degree zero, which means its value is simply a fixed number. In its general form, we write it as $P(x)=c$, where $c$ is a constant (any real number). This means that for any input value of $x$, the output of the polynomial is always the same, unchanging value, $c$.

Think of it like a soda machine that is stuck and only dispenses one type of drink, no matter which button you press. You can press the button for cola, lemon-lime, or root beer, but you always get the same orange soda. In this analogy, the button you press is the input $x$, and the orange soda is the constant output $c$.

Here are a few examples of constant polynomials:

• $f(x)=5$
• $g(x)=-17$
• $h(t)=\frac{3}{4}$
• $P(y)=\pi$

Notice that in each case, there is no variable (like $x$ or $t$) on the right side of the equation that affects the outcome. If we evaluate $f(x)=5$ for $x=2$, the result is $5$. If we evaluate it for $x=-100$, the result is still $5$. The value is constant.

What Is the Degree of a Constant Polynomial?

The degree of a polynomial is the highest exponent of its variable. This might seem tricky for a constant polynomial because you don't see a variable written down. However, we can use a property of exponents to reveal it.

Any non-zero number raised to the power of $0$ is equal to $1$. So, $x^0=1$. We can rewrite any non-zero constant polynomial $P(x)=c$ as:

When written this way, $P(x)=c{x}^0$, we can clearly see that the highest (and only) exponent of the variable $x$ is $0$. Therefore, the degree of any non-zero constant polynomial is $0$.

The Special Case: The Zero Polynomial

There is one important exception: the zero polynomial, $P(x)=0$. What is its degree? If we try the same trick, we run into a problem. We could write:

• $0=0\cdot x^1$
• $0=0\cdot x^2$
• $0=0\cdot x^{500}$

Since $0$ times anything is $0$, there is no single highest exponent we can assign. Because of this ambiguity, the degree of the zero polynomial is typically considered to be undefined. In more advanced mathematics, it is sometimes defined as $-1$ or $-\mathrm{\infty }$, but for Algebra, 'undefined' is the correct and simplest answer.

How Do You Identify a Constant Polynomial?

Identifying a constant polynomial is usually straightforward. You are looking for an expression that simplifies down to a single number, with no variables affecting its value. A term like $3x$ is not constant because its value changes as $x$ changes. A term like $3$ is constant.

Here is a table comparing constant polynomials with other expressions:

How Do You Graph a Constant Polynomial?

The graph of a constant polynomial is one of the easiest to recognize and draw: it is always a perfectly horizontal line. Let's explore why.

Consider the function $y=4$, which is another way of writing $f(x)=4$. This equation tells us that for any value of $x$ we choose, the $y$-value is always $4$. Let's make a quick table of values:

• If $x=-5$, $y=4$. Point: $(-5,4)$
• If $x=-1$, $y=4$. Point: $(-1,4)$
• If $x=0$, $y=4$. Point: $(0,4)$
• If $x=3$, $y=4$. Point: $(3,4)$

If you plot these points on a coordinate plane, you will see they all line up to form a horizontal line that passes through the $y$-axis at the value of the constant. Because a horizontal line has no 'rise', its slope is always $0$, which is another key feature.

What Are the Key Properties of Constant Polynomials?

Constant polynomials have several distinct properties that set them apart from other types of polynomials. Understanding these is key to mastering the concept.

• Form: They are always in the form $f(x)=c$, where $c$ is a real number.
• Degree: The degree is $0$ for any non-zero constant polynomial (e.g., $f(x)=8$). The degree is undefined for the zero polynomial ($f(x)=0$).
• Graph: The graph is always a horizontal line.
• Slope: The slope of the graph is always $0$. This indicates there is no rate of change; the function's value never increases or decreases.
• Y-intercept: The line crosses the y-axis at the point $(0,c)$. The y-intercept is simply the constant value itself.
• Roots (or Zeros): Roots are the $x$-values where the graph crosses the x-axis (where $y=0$).
  • A non-zero constant polynomial, like $f(x)=5$, has no roots. Its horizontal line graph is parallel to the x-axis and will never touch it.
  • The zero polynomial, $f(x)=0$, has infinite roots. Its graph is the x-axis, so it touches it at every single point.
• Domain and Range: The domain (all possible $x$-inputs) is all real numbers. The range (all possible $y$-outputs) consists of only one value: the constant $c$.

How Do You Perform Operations with Constant Polynomials?

Performing arithmetic operations like addition, subtraction, and multiplication with constant polynomials is very simple. Since their values don't depend on $x$, you are essentially just doing arithmetic with numbers.

Let's say we have two constant polynomials: $f(x)=c_1$ and $g(x)=c_2$.

• Addition: $(f+g)(x)=f(x)+g(x)=c_1+c_2$. The result is a new constant polynomial.
• Subtraction: $(f-g)(x)=f(x)-g(x)=c_1-c_2$. The result is also a constant polynomial.
• Multiplication: $(f\cdot g)(x)=f(x)\cdot g(x)=c_1\cdot c_2$. Again, the result is a constant polynomial.

The key takeaway is that when you add, subtract, or multiply constant polynomials, the result is always another constant polynomial.

Common Mistakes When Working with Constant Polynomials

Constant polynomials are simple, but a few common misunderstandings can trip students up. Be sure to avoid these pitfalls:

1. Confusing $y=c$ with $x=c$. The graph of $y=3$ is a horizontal line, which is a function. The graph of $x=3$ is a vertical line, which is not a function. A constant polynomial always corresponds to a horizontal line.
2. Assuming the Degree of $P(x)=0$ is 0. This is the most common mistake regarding degree. Remember, any non-zero constant like $7$ has degree $0$, but the special zero polynomial $0$ has an undefined degree.
3. Thinking a Constant Isn't a Polynomial. Some students believe a polynomial must have a visible variable. This is incorrect. A constant polynomial is the simplest type of polynomial, just as a single-story building is still a type of building.
4. Incorrectly Identifying Roots. A student might see $f(x)=6$ and think the root is $6$. A root is an $x$-value that makes the function equal to zero. Since $f(x)=6$ can never equal zero, it has no roots. Don't confuse the constant value with a root.

Constant Polynomials: Quick Summary

Here are the most important points to remember about constant polynomials. Use this as a quick reference guide.

• Definition: A function whose output value is the same for every input value.
• General Form:
  $P(x)=c$
  (where $c$ is a real number)
• Degree: The degree is $0$ if $c\ne 0$. The degree is undefined if $c=0$.
• Graph: A horizontal line that passes through the y-axis at $(0,c)$.
• Slope: The slope of the graph is always $0$.
• Roots: A non-zero constant polynomial has no roots. The zero polynomial ($P(x)=0$) has infinite roots.