/*! 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 26 Find the zeros for each polynomi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 26

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. $$f(x)=3(x+5)(x+2)^{2}$$

Short Answer

Expert verified
The zeros of the polynomial \(3(x+5)(x+2)^2\) are \(x=-5\) and \(x=-2\). The zero -5 has multiplicity 1 and the graph crosses the x-axis at this point. The zero -2 has multiplicity 2 and the graph touches the x-axis and turns around at this point.

Step by step solution

01

Identify The Zeros

The zeros of the function are identified as the values of \(x\) which makes \(f(x)=0\). In the polynomial \(3(x+5)(x+2)^2\), set each factor equal to zero and solve for \(x\). So we have \(x+5=0\) results to \(x=-5\), and \(x+2=0\) results to \(x=-2\). These are the zeros of the polynomial.
02

Determine The Multiplicity

The multiplicity of a zero is determined by the exponent of its corresponding factor in the function. The zero -5 has a corresponding factor of \(x+5\), which has an exponent of 1, so its multiplicity is 1. The zero -2 has a corresponding factor of \((x+2)^2\), which has an exponent of 2, so its multiplicity is 2.
03

Determine if the Graph Crosses or Touches the \(x\)-Axis

If the multiplicity of a zero is odd, then the graph crosses the \(x\)-axis at that zero. If the multiplicity of a zero is even, then the graph touches the \(x\)-axis and turns around at that zero. So, for \(x=-5\), the graph crosses the \(x\)-axis because its multiplicity is 1. For \(x=-2\), the graph touches the \(x\)-axis and turns around because its multiplicity is 2.

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

What is a polynomial inequality?

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(m\) and the square of \(n\) and inversely as \(p\) \(y=15\) when \(m=2, n=1,\) and \(p=6 .\) Find \(y\) when \(m=3, n=4,\) and \(p=10\).

The heat loss of a glass window varies jointly as the window's area and the difference between the outside and inside temperatures. A window 3 feet wide by 6 feet long loses 1200 Btu per hour when the temperature outside is \(20^{\circ}\) colder than the temperature inside. Find the heat loss through a glass window that is 6 feet wide by 9 feet long when the temperature outside is \(10^{\circ}\) colder than the temperature inside.

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies jointly as \(y\) and \(z\) and inversely as the square root of \(w\).

Describe how to graph a rational function.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied 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.