Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Polynomial functions are fundamental building blocks in algebra defined by sums of monomials with nonnegative integer exponents. Their structure, expressed in standard form, allows us to identify key features such as the degree, leading coefficient, zeros, and multiplicity, all of which dictate the behavior of their graphs—including turning points and end behavior. Mastery of these concepts is crucial for graph interpretation, solving equations, and applying polynomial models to real-world data.

Learning Objectives

1

Identify and classify polynomial functions by writing them in standard form and determining their degree.

2

Explain the roles of coefficients, leading terms, and constant terms in polynomial functions.

3

Analyze polynomial functions by finding their zeros, including repeated zeros (multiplicity), and determining how these affect the graph.

4

Utilize transformations and end behavior analysis to accurately sketch the graph of a polynomial function.

5

Apply polynomial modeling techniques to real-world data and interpret graphical characteristics such as turning points.

Key Concepts

CONCEPT

DEFINITION

Polynomial Function

A function of the form f(x)=aₙxⁿ+aₙ₋₁xⁿ⁻¹+…+a₁x+a₀ where the coefficients aₙ, aₙ₋₁, …, a₀ are constants, n is a nonnegative integer, and x is the variable. Its domain is the set of all real numbers.

Degree

The highest exponent n in a polynomial function when written in standard form; it determines many of the graph’s properties including end behavior and maximum turning points.

Leading Coefficient

The coefficient aₙ of the highest power term aₙxⁿ. It influences the steepness and the end behavior of the polynomial's graph.

Zero (or Root)

A value r for which f(r)=0; graphically, these are the x-intercepts of the polynomial function.

Multiplicity

The number of times a factor (x - r) appears in the factored form of a polynomial. A zero with even multiplicity means the graph just touches the x-axis at that point, while an odd multiplicity means the graph crosses the x-axis.

Standard Form

A representation of a polynomial with terms arranged in descending order of degree.

Power Function

A specific type of polynomial function that consists of a single term, f(x)=axⁿ, where n is a nonnegative integer. Its graph’s behavior varies with the parity of n.

End Behavior

The characteristic behavior of the graph of a polynomial function as x approaches positive or negative infinity, largely determined by the leading term.

Example Problems

Example 1

The intercepts of the equation $9 x^{2}+4 y=36$ are_____. $(\mathrm{pp} .159-160)$

Example 2

Is the expression $4 x^{3}-3.6 x^{2}-\sqrt{2}$ a polynomial? If so, what is its degree? (pp. $39-47)$

Example 3

To graph $y=x^{2}-4,$ you would shift the graph of $y=x^{2}$_____ a distance of _____ units.$(p p \cdot 247-256)$

Example 4

Use a graphing utility to approximate (rounded to two decimal places) the local maximum value and local minimum value of $\left.f(x)=x^{3}-2 x^{2}-4 x+5, \text { for }-3<x<3 \text { . (p. } 229\right)$

Example 5

True or False The $x$ -intercepts of the graph of a function $y=f(x)$ are the real solutions of the equation $f(x)=0$. (pp. $215-217)$
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Step-by-Step Explanations

QUESTION

Determine whether f(x) = 5x³ - (1/4)x² - 9 is a polynomial function and state its degree.

STEP-BY-STEP ANSWER:

Step 1: Write the function in standard form. Here it is already arranged in descending order: 5x³ - (1/4)x² - 9.
Step 2: Check that each exponent of x is a nonnegative integer (3 for x³, 2 for x², and 0 for the constant term).
Step 3: The highest exponent is 3, so the degree is 3.
Final Answer: f(x) is a polynomial function of degree 3.

Identifying a Polynomial Function

QUESTION

Given a polynomial factored as f(x) = (x + 3)(x - 2)², determine the real zeros and their multiplicities.

STEP-BY-STEP ANSWER:

Step 1: Set each factor equal to zero. For x + 3 = 0, solve x = -3.
Step 2: For (x - 2)² = 0, solve x - 2 = 0, giving x = 2.
Step 3: Identify multiplicity: x = -3 appears once (multiplicity 1) and x = 2 appears twice (multiplicity 2).
Final Answer: The zeros are -3 with multiplicity 1 (graph crosses the axis) and 2 with multiplicity 2 (graph touches the axis).

Determining Zeros and Their Multiplicity

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Common Mistakes

  • Misidentifying a function as a polynomial when it includes terms with fractional or negative exponents.
  • Confusing the number of terms with the degree; the degree depends solely on the highest power of x, not on how many terms there are.
  • Failing to convert the function to standard form, obscuring the leading term and the degree.
  • Overlooking the concept of multiplicity, which affects whether the graph crosses or merely touches the x-axis at a zero.