/*! 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 59 Use synthetic division to show t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 59

Use synthetic division to show that \(x\) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. $$x^{3}-7 x+6=0, \quad x=2$$

Short Answer

Expert verified
Given \(x=2\) is a root for the original cubic equation through synthetic division, the polynomial simplifies to \(x^{2}+2x-3\), which can be factored to give \(x^{2}+3x-x-3 = 0\) or \((x+3)(x-1)=0\). Thus, the other two roots are \(x = -3\) and \(x = 1\). So, the completely factored polynomial is \((x-2)(x+3)(x-1)\), which verifies that \(x=2\), \(x=-3\) and \(x=1\) are all real solutions to the given polynomial equation.

Step by step solution

01

Synthetic Division

First, let's use synthetic division to confirm if \(x=2\) is indeed a root of the polynomial \(x^{3}-7x+6\). Write down the coefficients of the polynomial (1, 0, -7, 6), put the potential root (2) in a box or off to the side, then carry out the synthetic division process. The process is as follows: Bring down the leading coefficient (1 in this case), multiply this by the potential root (2), and add the result to the second coefficient (0). Write the result (2) underneath the line. Repeat this process for the remaining coefficients.
02

Checking Result

After going through the synthetic division process, the final result at the bottom should be 0. If it is not 0, then \(x=2\) is not a solution of the given cubic equation. However, in this case, the result should be 0, which means \(x=2\) is indeed a solution of the given equation.
03

Factoring the Polynomial

The other numbers obtained from the synthetic division (except the 0 denoting the remainder) represent the coefficients of the simplified polynomial, in this case, \(x^{2}+2x-3\). This second-degree polynomial can be factored by using the zero factor property.
04

Finding the Roots

Set \(x^{2}+2x-3 = 0\), and solve the equation by factoring. The roots of this equation, along with \(x=2\) (the root that had been provided initially), are all the solutions to the original cubic equation. Therefore, the factored form of the original equation can also be written in terms of these roots.

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

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}-x+6$$

Find all real zeros of the function. $$f(x)=4 x^{3}-3 x-1$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(-x)$$

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-2 x$$

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.