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Chapter 1: Problem 143

In the following exercises, identify the coefficient of each term. \(5 r^{2}\)

Short Answer

Expert verified
The coefficient is 5.

Step by step solution

01

- Understand the Term

The given term is \(5r^{2}\). Terms in algebra are composed of coefficients (numerical factors) and variables raised to a power. Here, \(5r^{2}\) comprises a numerical part and a variable part.
02

- Identify the Coefficient

The coefficient of a term is the numerical factor that is multiplied by the variable(s). In \(5r^{2}\), the coefficient is the number \(5\) because it is the number directly multiplying the variable part \(r^{2}\).

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Key Concepts

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

coefficient identification
In algebra, identifying the coefficient is a fundamental skill. A coefficient is the numerical factor attached to the variable(s) in a term. The term can be a mix of numbers, variables, and exponents. For instance, in the term \(5r^{2}\), the coefficient is \(5\). This is because \(5\) is the numerical part multiplying the variable component \(r^{2}\). Simply put, the coefficient can be seen as the 'number' part of the term. To identify it, look for the quantity directly multiplied by the variables, regardless of what powers those variables have.
  • Example: In the term \(7xy^{3}\), the coefficient is \(7\).
  • Example: In the term \(-3a^{2}b\), the coefficient is \(-3\).
  • Remember: The coefficient always includes its sign, positive (+) or negative (-).
Understanding coefficients is essential because it helps simplify and solve algebraic expressions and equations.
numerical factors
Numerical factors are the numbers in algebraic expressions that multiply the variables. They play a crucial role in determining the overall value of each term. For instance, in our example term \(5r^{2}\), the numerical factor is \(5\). This means that no matter what value \(r\) has, it will be multiplied by \(5\).

Numerical factors are important because they:
  • Help scale the value of variables and expressions.
  • Allow us to combine like terms correctly.
  • Provide a clearer understanding of how numbers and variables interact.
By recognizing the numerical factors in an expression, you can better understand and solve algebraic problems. Always identify them clearly to ensure accuracy in calculations and problem-solving.
variables and exponents
Variables and exponents are the heart of algebraic expressions. Variables represent unknown values and are usually denoted by letters such as \(r\), \(x\), or \(y\). Exponents, on the other hand, indicate how many times to multiply the variable by itself.

Here's a breakdown:
  • In the term \(r\), \(r\) is the variable.
  • In the term \(r^{2}\), \(r\) is still the variable, but \(2\) is the exponent, indicating \(r\) is to be multiplied by itself once (i.e., \(r \cdot r\)).
Understanding variables and exponents is crucial for:
  • Solving algebraic equations.
  • Understanding polynomial expressions.
  • Tackling more complex mathematical concepts.
In our example term \(5r^{2}\), \(r\) is the variable and \(2\) is the exponent, showing the variable is squared. By mastering variables and exponents, you create a strong foundation for all future algebraic concepts and calculations.

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