/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Find the zeros for each polynomi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 27

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=4(x-3)(x+6)^{3}$$

Short Answer

Expert verified
The zeros of the function are \(x=3\), with a multiplicity of 1, and \(x=-6\), with a multiplicity of 3. At both zeros, the graph crosses the \(x\)-axis, since multiplicity for both is odd.

Step by step solution

01

Find the Zeros

To find the zeros of the function, we set \(f(x)\) equal to zero and solve for \(x\):\[4(x-3)(x+6)^{3}=0\]All factors of polynomial set to zero and get two solutions \(x=3\) and \(x=-6\].
02

Determine Multiplicities

The multiplicity of a zero is the power of its corresponding factor. Here, \(x=3\) comes from the factor \((x-3)\), and it appears only once, so its multiplicity is 1. Similarly, \(x=-6\) comes from the factor \((x+6)^3\), which appears three times, so its multiplicity is 3.
03

Describe the Behavior at Each Zero

At \(x=3\), since the multiplicity is odd (1), the graph crosses the \(x\)-axis. At \(x=-6\), since the multiplicity is odd (3), the graph also crosses the \(x\)-axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same vicwing rectangle. For odd values of \(n\), how does changing \(n\) affect the graph of \(y=\frac{1}{x^{m}} ?\)

What is a rational inequality?

Galileo's telescope brought about revolutionary changes in astronomy. A comparable leap in our ability to observe the universe took place as a result of the Hubble Space Telescope. The space telescope was able to see stars and galaxies whose brightness is \(\frac{1}{50}\) of the faintest objects observable using ground-based telescopes. Use the fact that the brightness of a point source, such as a star, varies inversely as the square of its distance from an observer to show that the space telescope was able to see about seven times farther than a ground-based telescope.

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

Use point plotting to graph \(f(x)=2^{x}\). Begin by setting up a partial table of coordinates, selecting integers from \(-3\) to 3 inclusive, for \(x\). Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.