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Chapter 6: Problem 3

Classify the following equation as linear or nonlinear: \(2 x=x^{3}-4\).

Short Answer

Expert verified
The equation is nonlinear because its degree is 3.

Step by step solution

01

Review the Equation

Look at the given equation: \(2x = x^3 - 4\). This equation is already set up in terms of one variable, \(x\).
02

Rearrange the Equation

To analyze the equation, rearrange it to get all terms on one side: \(x^3 - 2x - 4 = 0\). This presents the equation in standard form.
03

Identify the Degree of the Equation

In the rearranged equation \(x^3 - 2x - 4 = 0\), the highest power of \(x\) is 3. The degree of an equation is determined by the highest exponent of \(x\) present.
04

Classify Based on Degree

A linear equation has a degree of 1, while a nonlinear equation has a degree greater than 1. Since the degree of this equation is 3, it is not linear.

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Key Concepts

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

Degree of an Equation
The degree of an equation is a key concept in mathematics and is determined by the highest power of the variable present in the equation. In the example equation, \(x^3 - 2x - 4 = 0\), the degree is 3 because the highest exponent of \(x\) is 3. This tells us several things about the equation:
  • It involves higher-order terms, making it more complex than a simple linear equation.
  • A degree of 3 indicates that the graph of the equation will have more curvature, possibly crossing the x-axis up to three times.
The degree helps us classify equations into different types, such as linear, quadratic, cubic, etc. Understanding the degree is essential for determining the methods needed for solving such equations.
Linear Equations
Linear equations are the simplest types of equations that you鈥檒l encounter. In these equations, the degree is always 1, meaning the variable \(x\) is raised only to the power of 1. Linear equations take the form \(ax + b = 0\), where \(a\) and \(b\) are constants. Here's why linear equations are so straightforward:
  • Their graphs form straight lines when plotted on a coordinate plane.
  • They have a constant rate of change, making them predictable and easy to work with.
  • Solutions can be found using basic algebraic manipulations like addition, subtraction, or factoring.
In contrast, our original equation \(2x = x^3 - 4\) is not linear due to its degree being 3, which leads to more complex characteristics than just a simple line.
Polynomials
Polynomials are mathematical expressions consisting of variables, coefficients, and exponents. They can have multiple terms, each consisting of one variable raised to a certain power. For instance, the polynomial \(x^3 - 2x - 4\) consists of three terms:
  • \(x^3\) is the cubic term with a coefficient of 1.
  • \(-2x\) is the linear term with a coefficient of -2.
  • \(-4\) is the constant term.
Polynomials are classified based on their degree, as the degree affects both the shape of the graph and the methods used for solving them. Higher degree polynomials, like cubic and quartic, often require specialized techniques or numerical methods to find their roots.
Understanding polynomials and their properties is crucial for tackling complex equations and diverse mathematical problems.

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Most popular questions from this chapter

Factor completely: \(8 x^{2}+14 x y-15 y^{2}\).

Factor completely: \(-4 x^{7}+64 x^{3}\).

Factor: \(5 x^{2}-35 x-40\).

State the greatest common divisor of 36 and 60 .

Factor: \(-4 x^{3}-24 x^{2}-36 x\).
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