Calculus How To

Polynomial Function: Definition, Examples, Degrees

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Polynomial Function—Contents:

  1. Polynomial Function Definition
  2. Degrees:
  3. Limits for Polynomial Functions

See also: Leading Coefficients.

What is a Polynomial Function?

polynomial function

A polynomial function is made up of terms called monomials; If the expression has exactly two monomials it’s called a binomial. The terms can be:

  • Constants, like 3 or 523..
  • Variables, like a, x, or z,
  • A combination of numbers and variables like 88x or 7xyz.

polynomial function
You can’t have:

  • Fractional exponents, like x½
  • Negative exponents, like x-2
  • Variables within the radical (square root) sign. For example, √2.
  • Division by a variable.
  • An infinite number of terms.

Domain and Range of a Polynomial

The domain and range depends on the degree of the polynomial and the sign of the leading coefficient. Use the following flowchart to determine the range and domain for any polynomial function.
polynomial function domain range

Watch the short video for an explanation:

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Univariate Polynomial

A univariate polynomial has one variable—usually x or t. For example, P(x) = 4x2 + 2x – 9.In common usage, they are sometimes just called “polynomials”.

For real-valued polynomials, the general form is:

p(x) = pnxn + pn-1xn-1 + … + p1x + p0.

The univariate polynomial is called a monic polynomial if pn ≠ 0 and it is normalized to pn = 1 (Parillo, 2006). In other words, the nonzero coefficient of highest degree is equal to 1.

Zernike Polynomials

Zernike polynomials are sets of orthonormal functions that describe optical aberrations; Sometimes these polynomials describe the whole aberration and sometimes they describe a part. For example, “myopia with astigmatism” could be described as ρ cos 2(θ). This description doesn’t quantify the aberration: in order to so that, you would need the complete Rx, which describes both the aberration and its magnitude. Different polynomials can be added together to describe multiple aberrations of the eye (Jagerman, 2007).

Zernike polynomials aren’t the only way to describe abberations: Seidel polynomials can do the same thing, but they are not as easy to work with and are less reliable than Zernike polynomials.

Chebyshev Polynomials

Chebyshev polynomials crop up in many areas of calculus, including numerical integration, orthogonal polynomials and spectral methods for partial differential equations. They can also be used for curve fitting (finding a function that models a curve), interpolation and in multiple other areas of numerical analysis.

The general formula for a Chebyshev polynomial, for an integer n ≥ 0, is:

Tn(x) = cos(n cos-1x) ; -1 ≤ x ≤ 1


For n ≥ 2 (Smith, 2011):

Chebyshev Polynomials of the First Kind

Some authors refer to Chebyshev polynomials as just the Chebyshev polynomial of the first kind (Tn)—a polynomial in x of degree n, defined by the relation (Mason & Handscomb, 2002):
Tn(x) = cos nθ when x = cosθ.

The following table (Culham, 2020) lists the first 12 Chebyshev Polynomials of the first kind, obtained from Rodrigue’s formula:
rodrigues formula

T0(x) = 1
T1(x) = x
T2(x) = 2x2 – 1
T3(x) = 4x3 – 3x
T4(x) = 8x4 – 8x2 + 1
T5(x) = 16x5 – 20x3 + 5x
T6(x) = 32x6 – 48x4 + 18x2 – 1
T7(x) = 64x7 – 112x5 + 56x3 – 7x
T8(x) = 128x8 – 256x6 + 160x4 – 32x2 + 1
T9(x) = 256x9 – 576x7 + 432x5 – 120x3 + 9x
T10(x) = 512x10 – 1280x8 + 1120x6 – 400x4 + 50x2 – 1
T11(x) = 1024x11 – 2816x9 + 2616x7 – 1232x5 + 220x3 – 11x

Degrees of a Polynomial Function

“Degrees of a polynomial” refers to the highest degree of each term. To find the degree of a polynomial:

  1. Add up the values for the exponents for each individual term.
  2. Choose the sum with the highest degree.

degree of a polynomial

Example of a polynomial with 11 degrees.

First Degree Polynomial Function

First degree polynomials have terms with a maximum degree of 1. In other words, you wouldn’t usually find any exponents in the terms of a first degree polynomial. For example, the following are first degree polynomials:

  • 2x + 1,
  • xyz + 50,
  • 10a + 4b + 20.

The shape of the graph of a first degree polynomial is a straight line (although note that the line can’t be horizontal or vertical). The linear function f(x) = mx + b is an example of a first degree polynomial.

linear function

The graph of the polynomial function y =3x+2 is a straight line.

First degree polynomials have the following additional characteristics:

Second Degree Polynomial Function

Second degree polynomials have at least one second degree term in the expression (e.g. 2x2, a2, xyz2). There are no higher terms (like x3 or abc5). The quadratic function f(x) = ax2 + bx + c is an example of a second degree polynomial.

The graphs of second degree polynomials have one fundamental shape: a curve that either looks like a cup (U), or an upside down cup that looks like a cap (∩).

Graph of the second degree polynomial

Graph of the second degree polynomial 2x2 + 2x + 1.

Second degree polynomials have these additional features:

  • One extreme value (the vertex). A line of symmetry through the vertex.
  • Zero inflection points.
  • They take three points to construct; Unlike the first degree polynomial, the three points do not lie on the same plane.
  • Up to 2 roots.

Third Degree Polynomial

A cubic function (or third-degree polynomial) can be written as:
cubic function

where a, b, c, and d are constant terms, and a is nonzero.

Unlike quadratic functions, which always are graphed as parabolas, cubic functions take on several different shapes. We can figure out the shape if we know how many roots, critical points and inflection points the function has.

Cubic Function With Three Real Roots

A cubic function with three roots (places where it crosses the x-axis).

Third degree polynomials have been studied for a long time. In fact, Babylonian cuneiform tablets have tables for calculating cubes and cube roots. Chinese and Greek scholars also puzzled over cubic functions, and later mathematicians built upon their work.

Roots and Critical Points of a Cubic Function

Let’s suppose you have a cubic function f(x) and set f(x) = 0. Together, they form a cubic equation:

cubic equation

The solutions of this equation are called the roots of the polynomial. There can be up to three real roots; if a, b, c, and d are all real numbers, the function has at least one real root.

The critical points of the function are at points where the first derivative is zero:
first derivative of cubic function

We can use the quadratic equation to solve this, and we’d get:

It’s actually the part of that expression within the square root sign that tells us what kind of critical points our function has. Suppose the expression inside the square root sign was positive. Then we’d know our cubic function has a local maximum and a local minimum.

If b2-3ac is 0, then the function would have just one critical point, which happens to also be an inflection point. An inflection point is a point where the function changes concavity.

What about if the expression inside the square root sign was less than zero? Then we have no critical points whatsoever, and our cubic function is a monotonic function.

Limits for Polynomial Functions

There’s more than one way to skin a cat, and there are multiple ways to find a limit for polynomial functions. This can be extremely confusing if you’re new to calculus. But the good news is—if one way doesn’t make sense to you (say, numerically), you can usually try another way (e.g. graphically).

You can find a limit for polynomial functions or radical functions in three main ways:

Graphical and numerical methods work for all types of functions; Click on the above links for a general overview of using those methods. All work well to find limits for polynomial functions (or radical functions) that are very simple. You might also be able to use direct substitution to find limits, which is a very easy method for simple functions; However, you can’t use that method if you have a complicated function (like f(x) + g(x)).

This next section walks you through finding limits algebraically using Properties of limits . Properties of limits are short cuts to finding limits. They give you rules—very specific ways to find a limit for a more complicated function. For example, you can find limits for functions that are added, subtracted, multiplied or divided together.

Limit for Polynomial functions (Algebraic Method)

Example problem: What is the limit at x = 2 for the function
f(x) = (x2 +√2x)?

Step 1: Look at the Properties of Limits rules and identify the rule that is related to the type of function you have. The function given in this question is a combination of a polynomial function ((x2) and a radical function ( √ 2x). It’s what’s called an additive function, f(x) + g(x). The rule that applies (found in the properties of limits list) is:
lim x→a [ f(x) ± g(x) ] = lim1 ± lim2

Step 2: Insert your function into the rule you identified in Step 1.
lim x→2 [ (x2 + √ 2x) ] = lim x→2 (x2) + lim x→2(√ 2x).

Step 3: Evaluate the limits for the parts of the function. If you’ve broken your function into parts, in most cases you can find the limit with direct substitution:
lim x→2 [ (x2 + √2x) ] = (22 + √2(2) = 4 + 2

Step 4: Perform the addition (or subtraction, or whatever the rule indicates):
lim x→2 [ (x2 + √2x) ] = 4 + 2 = 6
That’s it!
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Aufmann, R. et al. (2005). Intermediate Algebra: An Applied Approach. Cengage Learning.
Culham, J. (2020). Chebyshev Polynomials. Retrieved August 22, 2020 from:
Davidson, J. (1998). First Degree Polynomials. Retrieved 10/20/2018 from:
Iseri, Howard. Lecture Notes: Shapes of Cubic Functions. MA 1165 – Lecture 05. Retrieved from
Jagerman, L. (2007). Ophthalmologists, Meet Zernike and Fourier! Trafford Publishing.
Mason, J. & Handscomb, S. (2002). Chebyshev Polynomials. CRC Press.
Parillo, P. (2006). MIT 6.972 Algebraic techniques and semidefinite optimization. Retrieved September 26, 2020 from:
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