Book cover for Precalculus: A Right Triangle Approach

Precalculus: A Right Triangle Approach

Judith A. Beecher, Marvin L. Bittinger, David J. Ellenbogen, Judith A. Penna

ISBN #9780321783967

5th Edition

5,905 Questions

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25,202 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the fundamental aspects of polynomial functions. The leading term plays a crucial role in determining a function's end behavior, with degree and leading coefficient being pivotal. The zero product property allows us to find the x-intercepts or the zeros of the polynomial, and understanding the multiplicity of zeros helps explain whether the graph crosses or touches the x-axis. These insights are essential for correctly graphing polynomials and for applying polynomial models to real-world scenarios.

Learning Objectives

1

Explain how the leading term of a polynomial function determines its end behavior.

2

Identify properties of polynomials including degree, leading coefficient, and how these properties affect the shape of the graph.

3

Apply the zero product property to find the real zeros (x-intercepts) of polynomial functions.

4

Distinguish between even and odd multiplicities and understand their impact on the graph (crossing vs. tangency).

Key Concepts

CONCEPT

DEFINITION

Leading Term

The term in a polynomial with the highest degree that largely determines the end behavior of the function.

Leading Coefficient

The coefficient of the leading term; its sign and magnitude affect the direction and steepness of the graph as x approaches ±∞.

Degree

The highest power of the variable in a polynomial which indicates the maximum number of x-intercepts and turning points.

End Behavior

The behavior of the graph of a polynomial function as x becomes very large (→ +∞ or -∞), determined by the leading term.

Zero Product Property

A principle stating that if a product of factors equals zero, then at least one of the factors must equal zero. Used to find the zeros of a polynomial once factored.

Multiplicity

The number of times a particular zero occurs. Zeros with even multiplicity cause the graph to touch and rebound from the x-axis; odd multiplicity causes the graph to cross the x-axis.

x-intercept

The point(s) where a polynomial function crosses the x-axis, corresponding to the real zeros of the function.

Example Problems

Example 1

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. $$ g(x)=\frac{1}{2} x^{3}-10 x+8 $$

Example 2

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. $$ f(x)=15 x^{2}-10+0.11 x^{4}-7 x^{3} $$

Example 3

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. $$ h(x)=0.9 x-0.13 $$

Example 4

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. $$ f(x)=-6 $$

Example 5

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. $$ g(x)=305 x^{4}+4021 $$
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Step-by-Step Explanations

QUESTION

Given a polynomial function f(x) = -x^6 + 3x^4 - 2x^2 + 5, determine its end behavior.

STEP-BY-STEP ANSWER:

Step 1: Identify the leading term. For f(x) = -x^6 + 3x^4 - 2x^2 + 5, the leading term is -x^6.
Step 2: Determine the degree of the polynomial, which is 6 (an even degree).
Step 3: Analyze the sign of the leading coefficient. Here it is negative (-1).
Step 4: Conclude the end behavior: For an even-degree polynomial with a negative leading coefficient, as x → ±∞, f(x) → -∞.
Final Answer: The graph falls to -∞ on both ends.

Leading-Term Test

QUESTION

Find the zeros of the polynomial function g(x) = (x - 2)(x + 1)^2.

STEP-BY-STEP ANSWER:

Step 1: Set the polynomial equal to zero: (x - 2)(x + 1)^2 = 0.
Step 2: Apply the zero product property. Set each factor equal to zero.
Step 3: Solve x - 2 = 0, which gives x = 2.
Step 4: Solve (x + 1)^2 = 0. Since the square of (x + 1) is zero, x + 1 = 0, giving x = -1.
Step 5: Identify the multiplicity: x = -1 has multiplicity 2 (even) and x = 2 has multiplicity 1 (odd).
Final Answer: The zeros are x = -1 (with even multiplicity) and x = 2 (with odd multiplicity).

Finding Zeros Using the Zero Product Property

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Common Mistakes

  • Confusing the leading coefficient with the constant term.
  • Neglecting the effect of even versus odd degree in determining end behavior.
  • Failing to account for multiplicity, leading to errors in predicting whether the graph crosses or just touches the x-axis.
  • Incorrectly applying the zero product property by not setting each factor equal to zero.