8.1: Polynomial Functions

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Objectives:

• Determine whether a given function is a polynomial function
• Identify the degree, leading term, and nonzero terms of a polynomial
• Use a graphing calculator to estimate certain characteristics of a given polynomial (Note: This no longer applies as of Spring 2018, as students won't necessarily have graphing calculators since we don't permit them on exams. But they should be able to do this given a graph.)
• Understand that the long-run behavior is the same for a polynomial its leading term

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Important Definitions

polynomial function, standard form, leading term, term, degree, long-run behavior, nonzero term

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Introduction: Where Students Will Struggle

• When explaining the definition of a polynomial, students will need to be reminded what a power function is.
• Some students will have trouble expanding the polynomials in Problem 1 and Problem 6a.
• Students find long-run behavior to be a challenging concept. This will be a major focus of the remainder of this chapter.
• Students have trouble keeping the definitions of degree, nonzero terms, and the leading term of a polynomial straight.
• Problem 7 is particularly challenging. A strategy to guide them with this problem is outlined in the lesson plan below.

Lesson Guide:

Warm-up

Have students do Problems 1 and 2. Some students may not remember how to expand the function given in Problem 1. This is a good opportunity to remind them how distributing works.

Identifying Polynomial Functions

Begin with the definition of a polynomial function:

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Definition: A polynomial function is a function that can be written as a sum of power functions whose exponents are non-negative integers.

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Give some examples and non-examples of polynomials to explain the definition. As some examples, you might use:

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Examples of polynomial functions:

• $p(x) = 7x^3 + x^2 + \pi$
• $q(x) = 2x^9$ (You could instead make this $2x^9+0x^2$ to set up the idea of nonzero terms.)
• Any quadratic function $r(x) = ax^2 + bx + c$.
• Any linear function $s(x) = mx + b$.

Examples of non-polynomial functions:

• $p(x) = x^\frac{1}{2}$
• $q(x) = x^{-1}$

(These are both power functions, but not polynomial functions.)

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Definition: For some polynomial function $f(x)$,

• Each individual power function $kx^p$ is called a term.
  • A term $kx^p$ is said to be nonzero if $k\neq 0$.
• The leading term is the term with the highest exponent for the variable.
• The degree is the largest exponent of $x$ that appears in the polynomial $f(x)$.
• The standard form of $f(x)$ is written as a sum of power functions, whose exponents decrease from left to right.

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Go back to the examples of polynomials already written on the board. Have students talk in their groups and try to decide the degrees and leading terms of the examples. After a few minutes, ask them what they came up with and write these next to the examples on the board.

Have students do Problem 3.

Using a Graphing Calculator to Estimate Characteristics of a Polynomial

Have students do Problems 4-5. Encourage them to use their graphing calculators (if they possess them) to help them graph the functions, and tell them that in this case, we are looking for the general shape of the function, not exact values (so their graphs don't need to be perfectly accurate). Ask students to write in their workbook what is similar and different about the functions in Problem 4 and Problem 5. After 10 or so minutes of working in groups, ask students to report out their findings. Ideally, they might see that:

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Student Observations:

• The polynomials have the same leading terms.
• The polynomials have behave the same way as $x \to \pm \infty$.

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Keeping the above two ideas in mind, graph the power function defined by the leading term of Problem 4 along with the functions in 4(a,b,c) to compare. Do the same with Problem 5. Use these graphs to show students that as we get far away from the origin, these polynomials behave like their leading terms.

Long-Run Behavior of a Polynomial

We have introduced long-run behavior previously, but you should re-introduce it in the context of polynomial functions.

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Definition: When viewed on a large enough scale, the graph of a polynomial function p(x) looks like the graph of its leading term. This behavior is called the long-run behavior of the polynomial.

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The terms long-run behavior and end behavior are used interchangeably. Let students know this. You may want to explain long-run behavior as being what a function does very far from the origin, or as $x \to \infty$ or $x \to -\infty$.

Do several examples to demonstrate this objective. As a suggestion:

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Example: For the following polynomials, state the degree, number of nonzero terms, and describe the long-run behavior.

• $p(x) = 16x^3 - 4x + 3x^2 + 10x^8$
• $g(x) = x^5 - x^2$
• $f(x) = x^2 (-x + 3)(2x^2 - 6) - 1$
• $h(x) = 1$

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Make sure to emphasize how they should be writing long-run behavior. This concept appears throughout the remainder of Chapter 8. For example, for p(x) in the above example, we want them to notice that p(x) behaves like its leading term, 10x^8, but additionally expect them to write "as $x \to \infty, p(x) \to \infty$ and as $x \to - \infty, p(x) \to \infty$." You might want to motivate this by asking them what happens when they plug very large positive or negative numbers into 10x^8.

Have students do Problems 6-8.

Problem 7 is a fairly challenging problem for students. A large part of this hurdle is likely the "proof-like" phrasing. Reframing the question in a less intimidating way will make them more likely to attempt it and less likely to give up. As a suggestion, you might ask students to read the problem on their own, then point out to the entire class that these are, at their core, just true/false questions. The only difference is that they need to explain why the statements are true or false. Do (a)(i) and (a)(ii) with them to show them how to explain a statement being true or produce a counterexample. Perhaps instruct them to go through the remainder of the problem initially, making their best guess as to whether statements are true or false. Then ask them to go back and fill in the details (justification).