- Polynomial Function Definition
- Limits for Polynomial Functions
See also: Leading Coefficients.
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.
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.
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.
Watch the short video for an explanation:
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:
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 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 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):
- Tn(x) is an n th-order polynomial in x.
- When n is an even integer, Tn(x) is an even function.
- When n is an odd integer, Tn(x) is an odd function.
- Tn(x) has n zeros in the open interval (-1, 1).
- Tn(x) has n + 1 extrema in the closed interval [-1, 1].
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:
|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” refers to the highest degree of each term. To find the degree of a polynomial:
- Add up the values for the exponents for each individual term.
- Choose the sum with the highest degree.
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.
First degree polynomials have the following additional characteristics:
- A single root, solvable with a rational equation.
- A constant rate of change with no extreme values or inflection points.
- The entire graph can be drawn with just two points (one at the beginning and one at the end).
- Symmetry for every point and line.
- The range is the set of all real numbers.
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 (∩).
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.
A cubic function (or third-degree polynomial) can be written as:
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.
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:
The critical points of the function are at points where the first derivative is zero:
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.
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.
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
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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: mhtl.uwaterloo.ca/courses/me755/web_chap6.pdf
Davidson, J. (1998). First Degree Polynomials. Retrieved 10/20/2018 from: https://www.sscc.edu/home/jdavidso/Math/Catalog/Polynomials/First/First.html
Iseri, Howard. Lecture Notes: Shapes of Cubic Functions. MA 1165 – Lecture 05. Retrieved from http://faculty.mansfield.edu/hiseri/Old%20Courses/SP2009/MA1165/1165L05.pdf
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: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/lecture-notes/lecture_05.pdf
Smith, J.O. Spectral Audio Signal Processing, http://ccrma.stanford.edu/~jos/sasp/, online book, 2011 edition, accessed August 23, 2020.