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Chapter 2: Problem 72

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Any problem that can be done by synthetic division can also be done by the method for long division of polynomials.

Short Answer

Expert verified
The statement is False. The true statement would be: 'Any problem that can be done by synthetic division can also be done by the method for long division of polynomials, but not all problems that can be done by long division of polynomials can be done by synthetic division.'

Step by step solution

01

Understanding the Context

Start with the knowledge about both methods. Synthetic division and long division of polynomials are methods used to divide polynomials. Synthetic division is a shortcut method which can be used only when dividing by a linear term (a polynomial of first degree) with a leading coefficient of 1. On the other hand, the long division of polynomials can be used to divide by any polynomial, irrespective of degree or the leading coefficient.
02

Comparing the two methods

Given the information for each method, it's clear that there are certain problems that can only be done by the method of long division of polynomials and not by synthetic division. Hence, the original statement isn't fully correct, as synthetic division has limitations that long division does not.
03

Adjusting the Statement

To make the statement true, it should be revised to: 'Any problem that can be done by synthetic division can also be done by the method for long division of polynomials, but not all problems that can be done by long division of polynomials can be done by synthetic division.'

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Key Concepts

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

Synthetic Division
Synthetic division is a streamlined technique to divide a polynomial by a linear term. This method is especially appreciated for its simplicity and speed, but it does come with certain restrictions.
It is primarily used when the divisor is a linear polynomial, which means the polynomial is of the first degree (like \( ax + b \), where \( a \) must equal 1).
When using synthetic division, we focus on the zero of the linear term. If the linear term is \( x - c \), the associated zero is \( c \). This zero helps simplify the division process significantly by reducing the calculation steps compared to long division.
Here are some advantages of synthetic division:
  • Simplifies the process with fewer steps compared to long division.
  • Quickly gives quotient and remainder.
  • Particularly effective for handling polynomials with a leading coefficient of 1.
However, it鈥檚 good to note that synthetic division is not applicable to situations where the divisor has a degree greater than 1 or when its leading coefficient isn鈥檛 1.
Long Division of Polynomials
Long division of polynomials is a more comprehensive method compared to synthetic division. This method works for any polynomial divisor, regardless of its degree or leading coefficient.
The process is similar to the long division of numbers, where you divide, multiply, subtract, and bring down terms one by one. Here's a quick breakdown of how it works:
  • Divide: The first term of the dividend by the first term of the divisor.
  • Multiply: The result is multiplied by the entire divisor.
  • Subtract: This product is then subtracted from the dividend.
  • Bring Down: The next term of the dividend is brought down, and the process repeats.
This procedure continues until you've gone through all the terms of the dividend, leaving you with a quotient and possibly a remainder.
Long division is particularly useful when dealing with higher degree polynomials or complex divisors and doesn't have the limitations associated with synthetic division.
Linear Term
In polynomial mathematics, a linear term refers to a polynomial of degree one, such as \( ax + b \). This term is a foundational element of polynomials because it represents the simplest type of polynomial division.
Linear terms are significant in both synthetic division and long division as they often act as divisors. With synthetic division, the linear term must have a leading coefficient of 1, simplifying calculations.
Understanding linear terms is crucial because:
  • They determine the type of division that can be applied effectively.
  • Affects the strategies for finding polynomial zeros, which are essential in simplification, factoring, and graphing of polynomial functions.
Grasping the concept of linear terms enables a better understanding of how and when to apply synthetic or long division, enhancing one鈥檚 ability to solve polynomial problems efficiently.

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Most popular questions from this chapter

An athlete whose event is the shot put releases the shot wilh the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of \(35^{\circ}\) and \(65^{\circ} .\) Exercises \(57-58\) are based on the functions that model the parabolic paths. (table cannot copy) When the shot whose path is shown by the red graph is released at an angle of \(65^{\circ},\) its height, \(g(x),\) in feet, can be modeled by $$ g(x)=-0.04 x^{2}+2.1 x+6.1 $$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the red graph. a. What is the maximum height, to the nearest tenth of a foot, of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

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