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4: Polynomial and Rational Functions. - Mathematics


  • 4.1: Quadratic Functions
    In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
  • 4.2: Polynomial Functions
    Suppose a certain species of bird thrives on a small island. The population can be estimated using a polynomial function. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.
  • 4.3: Rational Functions
    In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

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Is a polynomial function a rational function?

It's my first time on this forum so please forgive any errors in the presentation of my question.

My query refers to definitions of polynomials and rational functions. I have just started Ferrar's 'Higher Alegbra for Schools' and have become bogged down in the definitions. I have attached link but will try and explain myself first.

The first chapter begins with a number of definitions, in particular definitions of polynomial, rational and algebraic functions. Polynomials are defined strictly for positive exponents (n), whilst rational as the ratio of two polynomials. Mr Ferrar then goes on to describe and define algebraic functions. His example (see link) is a quadratic in $y$ with polynomials in $x$ . He then goes on to define, in general, that algebraic functions are the same as the example but with rational functions of $x$ .

Herein lies my lack of understanding. Put plainly my question is this,

Does a polynomial function imply rational?

By modern definitions it would appear so, but I am unable to see how Mr Ferrar's generality is to include polynomials as his definition seems inadequate.

Please forgive any ignorance on my part, I'm sure there's a subtlety here that I'm missing but I'm not quite sure what it is.


4: Polynomial and Rational Functions. - Mathematics

Students intending to pursue college majors in the mathematical, physical, and biological sciences and engineering should study this unit. Students intending to pursue programs in social, management, and some of the health sciences or humanities, may omit or reduce the amount of time spent studying this unit and instead study other units from the text. (See the descriptions of Course 4 Units.)

Unit Overview

Polynomial and Rational Functions extends student ability to use polynomial and rational functions to represent and solve problems from real-world situations while focusing on symbolic and graphical patterns.

  • To describe and use the concepts of zeroes and end behavior of functions in mathematical, scientific, and everyday situations
  • To use polynomial and rational functions to model data patterns
  • To describe and illustrate the relationship between the graph of a polynomial or rational function and its symbolic representation
  • To develop facility with manipulating and reasoning about polynomial and rational symbolic representations
  • To determine all complex number roots of polynomials and to add, subtract, multiply, and divide complex numbers

Sample Overview

There are two different samples from Polynomial and Rational Functions. The first sample is the first two investigations of Lesson 2 of the unit. In these investigations, students examine nested, standard, and factored forms of polynomials, the effects of the symbolic form on computational efficiency, and the information provided by each form. Students develop skill in factoring polynomials. This work builds the necessary knowledge to introduce complex numbers in the third investigation.

The second sample material is the "Looking Back" lesson for this unit. This lesson is intended to provide students with tasks that will encourage them to look back at the unit as a whole. Students review, synthesize, and apply the knowledge gained during the study of the unit.

Instructional Design

Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a common pattern as elaborated under Instructional Design.

View Sample Material

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Contact Adobe with any technical questions about their software or its installation.

How the Algebra and Functions Strand Continues

The next unit in Course 4, Functions and Symbolic Reasoning, extends student ability to manipulate symbolic representations of exponential, common and natural logarithmic, and trigonometric functions and to solve exponential, logarithmic, and trigonometric equations. Trigonometric identities are developed and proved or disproved. Geometric representations of complex numbers are used to reason about and to find roots of complex numbers. Algebraic representations of surfaces and conic sections are introduced in Unit 8, Space Geometry.

A unit that develops understanding and skill in the use of standard spreadsheet operations while reviewing and extending many of the basic algebra topics from Courses 1-3 is included for students intending to pursue college programs in social, management, and some of the health sciences or humanities.

Copyright 2019 Core-Plus Mathematics Project. All rights reserved.


Short run Behavior: Intercepts

As with any function, we can find the vertical intercepts of a quadratic by evaluating the function at an input of zero, and we can find the horizontal intercepts by solving for when the output will be zero. Notice that depending upon the location of the graph, we might have zero, one, or two horizontal intercepts.

zero horizontal intercepts one horizontal intercept two horizontal intercepts

We can determine the vertical and horizontal intercepts of a quadratic using the quadratic formula.

Quadratic formula: for a quadratic function given in standard form , the quadratic formula gives the horizontal intercepts of the graph of this function.

A ball is thrown upwards from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation

.

What is the maximum height of the ball?

When does the ball hit the ground?

To find the maximum height of the ball, we would need to know the vertex of the quadratic.

The ball reaches a maximum height of 140 feet after 2.5 seconds.

To find when the ball hits the ground, we need to determine when the height is zero – when H(t) = 0. While we could do this using the transformation form of the quadratic, we can also use the quadratic formula:

Since the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions:

or

The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds.

The supply for a certain product can be modeled by and the demand can be modeled by , where p is the price in dollars, and q is the quantity in thousands of items. Find the equilibrium price and quantity.

Recall that the equilibrium price and quantity is found by finding where the supply and demand curve intersect. We can find that by setting the equations equal:

Add 2q 2 to both sides

Divide by 5 on both sides

Take the square root of both sides

Since it doesn’t make sense to talk about negative quantities, the equilibrium quantity is q = 18. To find the equilibrium price, we evaluate either function at the equilibrium quantity.

The equilibrium is 18 thousand items, at a price of 遬.


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Polynomial and Rational Functions - PowerPoint PPT Presentation

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  1. For (f(x) = x^3 - 7x^2 + 7x + 15), the possible rational roots are: [left < pm 1, pm 3, pm 5, pm 15 ight >]
  2. For (f(x) = x^4 - 4x^3 - 13x^2 + 4x + 12), the possible rational roots are: [left < pm 1, , pm 2, pm 3, pm 4, pm 6, pm 12 ight >]
  3. For (f(x) = x^5 - 3x^4 + 7x^2 + 10), the possible rational roots are: [left < pm 1, pm 2, pm 5, pm 10 ight >]
  4. For (f(x) = 2x^3 - 6x^2 + 5x - 8), the possible rational roots are: [left < pm frac<1><2>, pm 1, pm 2, pm 4, pm 8 ight >]
  5. For (f(x) = 6x^3 - 11x^2 - 7x + 10), the possible rational roots are: [left < pm frac<1><6>, pm frac<1><3>, pm frac<1><2>, pm frac<2><3>, pm frac<5><6>, pm 1, pm frac<5><3>, pm 2, pm frac<5><2>, pm frac<10><3>, pm 5, pm 10 ight >]
  6. For (f(x) = 3x^6 - 4x^5 + 2x^4 - 3x + 12), the possible rational roots are: [left < pm frac<1><3>, pm frac<2><3>, pm 1, pm frac<4><3>, pm 2, pm 5, pm 10 ight >]
  7. For (f(x) = -2x^3 + 9x^2 +6x - 5), the possible rational roots are: [left < pm frac<1><2>, pm 1, pm frac<5><2>, pm 5 ight >]
  8. For (f(x) = 4x^4 - 20x^3 - 60x^2 + 20x + 56), the possible rational roots are: [left < pm frac<1><4>, pm frac<1><2>, pm 1, pm 2, pm 4, pm 8, pm 14, pm 28, pm 56 ight >]
  • Step 1: use the rational root theorem to list all of the polynomial's potential zeros.
  • Step 2: use "trial and error" to find out if any of the rational numbers, listed in step 1, are indeed zero of the polynomial.

The following two tutorials illustrate how the rational root theorem can be used to help find a polynomial's zeros.

Tutorial: Rational Root Theorem

In this tutorial, we find the zeros of a polynomial using the rational root theorem. The polynomial is: [f(x) = x^3 - 2x^2 - 5x + 6]

Tutorial: Rational Root Theorem

In this tutorial, we find the zeros of a polynomial using the rational root theorem. The polynomial is: [f(x) = 2x^4 + x^3 - 13x^2 - 6x + 6]


4: Polynomial and Rational Functions. - Mathematics

Figure 1. 35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. (credit “film”: modification of work by Horia Varlan credit “memory cards”: modification of work by Paul Hudson)

Digital photography has dramatically changed the nature of photography. No longer is an image etched in the emulsion on a roll of film. Instead, nearly every aspect of recording and manipulating images is now governed by mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses complex polynomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications.


4: Polynomial and Rational Functions. - Mathematics

If modeling via polynomial models is inadequate due to any of the limitations above, you should consider a rational function model.

  1. Rational function models have a moderately simple form.
  2. Rational function models are a closed family. As with polynomial models, this means that rational function models are not dependent on the underlying metric.
  3. Rational function models can take on an extremely wide range of shapes, accommodating a much wider range of shapes than does the polynomial family.
  4. Rational function models have better interpolatory properties than polynomial models. Rational functions are typically smoother and less oscillatory than polynomial models.
  5. Rational functions have excellent extrapolatory powers. Rational functions can typically be tailored to model the function not only within the domain of the data, but also so as to be in agreement with theoretical/asymptotic behavior outside the domain of interest.
  6. Rational function models have excellent asymptotic properties. Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite (x) --> values. Thus, rational functions can easily be incorporated into a rational function model.
  7. Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator. This in turn means that fewer coefficients will be required compared to the polynomial model.
  8. Rational function models are moderately easy to handle computationally. Although they are nonlinear models, rational function models are a particularly easy nonlinear models to fit.
    Given that data has a certain shape, what values should be chosen for the degree of the numerator and the degree on the denominator?

    If the numerator and denominator are of the same degree ((n=m)), then (y = a_n / b_m) is a horizontal asymptote of the function.

Note: This type of fit, with the response variable appearing on both sides of the function, should only be used to obtain starting values for the nonlinear fit. The statistical properties of models like this are not well understood.


Unit 6 – Polynomials and Rational Functions

The complexity and algebra of polynomials is examined in the first two lessons of this unit. Particular care is given to linking the factors of a polynomial to its x-intercepts. Rational functions and inverse variation are briefly covered and give way to the traditional rational algebra that is so essential to master before moving to precalculus. Besides reviewing knowledge and skills from Algebra 1, the newer concepts of complex fractions and fractional equations are introduced.

Unit 6 Review – Polynomial and Rational Functions

Unit 6 – Mid-Unit Quiz (Through Lesson 5) – Form A

Unit 6 – Mid-Unit Quiz (Through Lesson 5) – Form B

Unit 6 – Polynomial Challenge (After Lesson 2)

Unit 6 – Polynomial Challenge Teacher Directions

Unit 6.Rational Puzzles Activity (After Lesson 7)

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Watch the video: The parts of polynomial expressions. Polynomial and rational functions. Algebra II. Khan Academy (October 2021).