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Chapter 4: Problem 5

Determine whether the terms are like or unlike. $$6 x^{2}, 6 x$$

Short Answer

Expert verified
The terms \(6x^2\) and \(6x\) are unlike terms.

Step by step solution

01

Identify Variables and Powers

First, Identify variables and powers in both terms. In \(6x^2\), the variable is \(x\) and the power is \(2\). In \(6x\), the variable is \(x\) and the power is \(1\) (implicit).
02

Compare Variables and Powers

Next, Compare the variables and their powers in the two terms. Here in \(6x^2\) the variable is \(x\) with power \(2\), while in \(6x\) the variable is \(x\) with power \(1\). Note that the variables are the same (\(x\) in both) but the powers are different (2 and 1). Therefore, the terms are not like terms.

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (such as addition or multiplication). They are used to represent quantities in general terms.
Typically, an algebraic expression might look like something such as \(3x + 4y - 5\). This expression contains:
  • Numerical coefficients (like 3 and 4)
  • Variables (\(x\) and \(y\))
  • Constants (such as -5)
Understanding how these elements interact is crucial for manipulating expressions and solving equations. In algebra, we often combine or simplify expressions by using operations and identifying like terms. Getting comfortable with the structure of algebraic expressions is essential for success in mathematics.
The Role of Variables in Algebra
Variables are symbols, commonly letters like \(x\), \(y\), or \(z\), that stand in for unknown values or quantities in mathematics. They are an essential part of algebraic expressions as they allow for generalization and abstraction.
Variables can represent:
  • Unknown numbers that we're trying to find
  • Quantities that can change (e.g., time, distance)
  • Constants, depending on the context of the problem
When working with algebraic expressions, identifying variables helps in manipulating and solving them. For example, in the expression \(6x^2 + 6x\), "\(x\)" is the variable. It allows us to perform operations algebraically rather than numerically, giving flexibility in problem-solving.
Grasping Powers of Variables
Powers of variables, also known as exponents, indicate how many times a variable is multiplied by itself. For instance, \(x^2\) means \(x\) multiplied by \(x\) (or \(x \times x\)).
They serve several purposes:
  • Determining the degree of polynomial terms
  • Influencing the behavior of functions (growth/decay rates)
  • Changing the characteristics of algebraic expressions
In our original exercise, the terms \(6x^2\) and \(6x\) have the same variable, "\(x\)," but different powers (2 and 1, respectively). This difference is crucial for identifying like terms, as like terms must have identical variables raised to the same power. Thus, it is important always to consider the powers when working with algebraic expressions to determine their relationship and how they can be combined.

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