/*! 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 49 Use synthetic division to determ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 49

Use synthetic division to determine whether the given number is a zero of the polynomial function. $$ 2-i ; \quad f(x)=x^{2}+3 x+4 $$

Short Answer

Expert verified
\(2 - i\) is not a zero of the polynomial function \(f(x) = x^2 + 3x + 4\).

Step by step solution

01

- Understand the Problem

The goal is to determine if the complex number \(2 - i\) is a zero of the polynomial function \(f(x) = x^2 + 3x + 4\). Synthetic division will be used to test this.
02

- Set Up Synthetic Division

Write the coefficients of the polynomial \(f(x)\). For \(f(x) = x^2 + 3x + 4\), the coefficients are \([1, 3, 4]\). We will use \(2 - i\) for the synthetic division.
03

- Perform Synthetic Division

Set up synthetic division by writing \(2 - i\) on the left, and the coefficients \([1, 3, 4]\) of the polynomial on the right side of the line. 1. Bring down the first coefficient (1).2. Multiply \(1\) by \(2 - i\), which results in \(2 - i\). Write this under the next coefficient (3).3. Add: \(3 + (2 - i) = 5 - i\). Write this sum underneath.4. Multiply \(5 - i\) by \(2 - i\), giving \((5 - i)(2 - i) = 10 - 5i - 2i + i^2 = 10 - 7i - 1 = 9 - 7i\).5. Add: \(4 + (9 - 7i) = 13 - 7i\).
04

- Analyze the Remainder

The final value after synthetic division is \(13 - 7i\). Since this is not zero, \(2 - i\) is not a zero of the polynomial function.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers in Polynomials
Polynomials often have complex numbers as their roots. A complex number is a number that includes a real part and an imaginary part, typically written as \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\). For example, in the problem, we are given the complex number \(2 - i\). When dealing with complex roots, always remember:
  • Every complex number has a real and an imaginary component.
  • Adding or multiplying complex numbers involves combining like terms.
  • Imaginary numbers follow unique rules, such as \(i^2 = -1\).
Working with complex numbers requires careful arithmetic, but following these rules will help ensure accurate calculations. In our context, we use them to test if they are zeroes of polynomial functions.
Zeroes of Polynomial Functions
Zeroes of polynomial functions are the values of \(x\) that make the function equal to zero. In other words, they are the solutions to \(f(x) = 0\). For example, if \(f(x) = x^2 + 3x + 4\), we want to determine if \(2 - i\) is a zero of this polynomial function.

Here's why zeroes are important:
  • They reveal where the graph of the polynomial crosses or touches the x-axis.
  • They are critical for factoring polynomials and simplifying expressions.
  • Finding zeroes can provide insights into the behavior of the function.
To check if a number is a zero of a polynomial, we substitute it into the polynomial and see if it makes the polynomial equal to zero. If it does, it's a zero; if it doesn't, it's not. In this exercise, we use synthetic division to perform this check. This method is especially useful for polynomials with complex roots.
Step-by-Step Synthetic Division
Synthetic division is a simplified method for dividing polynomials, especially useful when testing potential zeroes. Here鈥檚 the step-by-step guide, as applied in the problem:
Step 1 - Write the Coefficients
For \(f(x) = x^2 + 3x + 4\), list the coefficients: \([1, 3, 4]\).

Step 2 - Set Up the Division
Place the complex number \(2 - i\) outside the division setup and the coefficients inside.

Step 3 - Perform Synthetic Division
  1. Bring down the first coefficient (1).
  2. Multiply by \(2 - i\): \(1 \cdot (2 - i) = 2 - i\). Write this under the next coefficient (3).
  3. Add the results: \(3 + (2 - i) = 5 - i\).
  4. Multiply the sum by \(2 - i\): \((5 - i)(2 - i) = 10 - 5i - 2i + i^2 = 10 - 7i - 1 = 9 - 7i\).
  5. Add the multiplication result to the next coefficient: \(4 + (9 - 7i) = 13 - 7i\).

Step 4 - Analyze the Remainder
The final result is the remainder: \(13 - 7i\). Since the remainder is not zero, \(2 - i\) is not a zero of the polynomial. Always ensure you correctly perform each arithmetic step, especially with complex numbers, to avoid mistakes.

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

Graph each rational function. $$f(x)=\frac{3}{(x+4)^{2}}$$

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier sections. What are the \(x\) -intercepts of the graph of \(f ?\)

Approximate to the nearest hundredth the coordinates of the turning point in the given interval of the graph of each polynomial function. \(f(x)=x^{3}+4 x^{2}-8 x-8, \quad[-3.8,-3]\)

Approximate all real zeros of each function to the nearest hundredth. \(f(x)=0.86 x^{3}-5.24 x^{2}+3.5 x+7.8\)

For each polynomial function, use the remainder theorem and synthetic division to find \(f(k) .\) $$ f(x)=2 x^{3}-3 x^{2}-5 x+4 ; \quad k=2 $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.