/*! 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 19 Find the rational zeros of the f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 19

Find the rational zeros of the function. $$f(x)=x^{3}-7 x-6$$

Short Answer

Expert verified
The rational zeros of the function \(f(x)=x^{3}-7x-6\) are \(x = -2, 3\)

Step by step solution

01

Identify the constant and leading coefficient

In our polynomial \(f(x)=x^{3}-7x-6\), the constant term (the term without a variable) is -6 and the leading coefficient (the coefficient of the term with the highest exponent) is 1.
02

List factors

List all factors of the constant term -6 and the leading coefficient 1. Factors of -6 are \(\pm1, \pm2, \pm3, \pm6\) and factors of 1 are \(\pm1\).
03

Determine possible rational zeros

Possible rational zeros can be obtained by dividing each factor of the constant term by each factor of the leading coefficient. As the factors of the leading coefficient is \(\pm1\), in this case, the factors of the constant term will remain the same. So, the possible rational zeros are \(\pm1, \pm2, \pm3, \pm6\).
04

Test possible zeros

The next step is to substitute each possible zero into the polynomial and determine if it makes the polynomial equal to zero. Check this for \(x = \pm1, \pm2, \pm3, \pm6\). After performing the calculations, it is found that \(x = 1\), \(x = -2\), and \(x = 3\) make the polynomial equal to zero (those are valid zeros of the function). When substituting, for example, \(x = 1\) we get \(f(1) = (1)^3 - 7*(1) - 6 = -12\, which is not equal to zero, so \(x = 1\) is not a root.
05

Present the solution

The rational zeros of the function \(f(x)=x^{3}-7x-6\) are \(x = -2\) and \(x = 3\).

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

Decide whether the statement is true or false. Justify your answer. If \(x=-i\) is a zero of the function \(f(x)=x^{3}+i x^{2}+i x-1\) then \(x=i\) must also be a zero of \(f\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}-3 x^{2}+4 x-2$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}+9 x^{2}+27 x+35$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.