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Chapter 2: Problem 17

In Exercises 11-18, identify the coefficient of the term. $$ 2 \pi x^{2} $$

Short Answer

Expert verified
The coefficient of the term \(2 \pi x^{2}\) is \(2 \pi\).

Step by step solution

01

Understand the term

The given term is \(2 \pi x^{2}\). This is an algebraic term so it might have a coefficient.
02

Identify the coefficient

The coefficient is the numerical part of the term that is multiplying the variable, here it is \(2 \pi\).

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Key Concepts

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

Algebraic Terms
In algebra, terms are the building blocks of mathematical expressions. A term might consist of numbers, variables, or both, combined through multiplication or division.
An algebraic term such as \(2 \pi x^{2}\) in our exercise is a combination of numbers (\(2\) and \(\pi\)) and a variable part (\(x^{2}\)).
These elements are usually multiplied together to form a single expression or term.
  • **Numbers**: These could be constants like \(2\), or a constant value like \(\pi\), which is about 3.14159.
  • **Variables**: Typically represented by letters like \(x\), they are symbols that can take various numerical values.
  • **Power of a Variable**: In the term \(x^{2}\), the variable \(x\) is raised to the power of 2, indicating it's multiplied by itself.
Recognizing how these components work together is crucial, as it paves the way to understanding more complex algebraic expressions.
Numerical Coefficient
The numerical coefficient in an algebraic term is the part that consists solely of numbers. It dictates the value by which the variable is influenced in the expression.
For the term \(2 \pi x^{2}\), the numerical coefficient is \(2 \pi\). This tells us how many of the \(x^{2}\) we have in this specific expression.
Identifying the numerical coefficient involves looking at the multiplicative parts of a term that are not the variable. Here's how you can spot it:
  • Focus on the numbers: In \(2 \pi x^{2}\), \(2 \pi\) is purely numerical.
  • Include constants: \(\pi\) is a constant, so treat it as part of the coefficient here.
Understanding numerical coefficients helps to quickly simplify, add, or compare algebraic expressions by focusing on the numerical impact of each term.
Variable Identification
Variables in algebra are typically used to represent unknowns or values that can change. Identifying variables in an expression helps in analyzing how changes in these unknowns affect the overall expression.
For the term \(2 \pi x^{2}\), the variable here is \(x\). It's critical to recognize not just the variable itself but its power or exponent as well.
Here are some tips for identifying variables:
  • Look for letters: Common variables are \(x, y, z\).
  • Consider their exponents: In high-level terms, \(x^2\) implies that changes in \(x\) have a squared effect on the expression.
Understanding variables allows you to manipulate algebraic expressions and predict how changes in one variable might affect the result. It forms a basis for solving equations and in modeling real-world situations.

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