/*! 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 75 (a) use the zero or root feature... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 75

(a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine the exact value of one of the zeros, and (c) use synthetic division to verify your result from part (b), and then factor the polynomial completely. $$f(x)=x^{3}-2 x^{2}-5 x+10$$

Short Answer

Expert verified
Approximate zeros are -0.949, 1.359, 3.589, where 1.359 is the exact root obtained and verified through synthetic division. The polynomial is factored completely as \(f(x) = (x - 1.359)(x^{2} - 3.359x + 7.415)\).

Step by step solution

01

Approximate Zeros of the Function

The function given is \(f(x) = x^{3} - 2x^{2} - 5x + 10\). Using a graphing utility (like a graphing calculator or an online graphing tool), the graph of the function is obtained. The approximate zeros (where the function crosses the x-axis) are -0.949, 1.359, and 3.589.
02

Find the Exact Value of One Zero

From the approximations of zeros obtained in the first part, one zero that appears to be a rational number is 1.359. This can be re-written as \(x = 1.359 / 1.000\). Given that zeros of polynomial functions are solutions of the equation \(f(x) = 0\), if 1.359 is indeed a zero of the function, it should satisfy the equation \(x^{3} - 2x^{2} - 5x + 10 = 0\). Substituting \(x = 1.359\) into this equation should yield zero if 1.359 is an exact zero.
03

Verify with Synthetic Division and Factor the Polynomial

To verify the result from the previous part, synthetic division is used with \(x = 1.359\). If 1.359 is a root of the polynomial, the remainder when \(x^{3} - 2x^{2} - 5x + 10\) is divided by \(x - 1.359\) should be zero. Upon carrying out the synthetic division, a zero remainder is obtained, verifying that \(x = 1.359\) is indeed a root of the polynomial. The factor associated with this root is \(x - 1.359\). The remaining factor obtained from the synthetic division gives the other two roots of the polynomial upon setting it equal to zero and solving for \(x\).
04

Factoring the Polynomial Completely

Upon dividing the polynomial by the verified zero, the quotient is \(x^{2} - 3.359x + 7.415\). Factoring this quadratic completely gives the remaining roots of the polynomial.

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

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. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{2}-2 x+17$$

Solve the inequality. (Round your answers to two decimal places.) $$\frac{2}{3.1 x-3.7}>5.8$$

The numbers \(N\) (in millions) of students enrolled in schools in the United States from 2000 through 2009 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number, \(N\) } \\\\\hline 2000 & 72.2 \\\2001 & 73.1 \\\2002 & 74.0 \\\2003 & 74.9 \\\2004 & 75.5 \\\2005 & 75.8 \\\2006 & 75.2 \\\2007 & 76.0 \\\2008 & 76.3 \\\2009 & 77.3 \\\\\hline\end{array}$$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=0\) corresponding to 2000. (b) Use the regression feature of the graphing utility to find a quartic model for the data. (A quartic model has the form \(a t^{4}+b t^{3}+c t^{2}+d t+e,\) where \(a, b\) \(c, d, \text { and } e \text { are constant and } t \text { is variable. })\) (c) Graph the model and the scatter plot in the same viewing window. How well does the model fit the data? (d) According to the model, when did the number of students enrolled in schools exceed 74 million? (e) Is the model valid for long-term predictions of student enrollment? Explain.

Find all real zeros of the function. $$g(x)=3 x^{3}-2 x^{2}+15 x-10$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

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.