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Chapter 3: Problem 9

List the terms and identify each term as a variable term or a constant term. $$2 a+5 b^{2}+6$$

Short Answer

Expert verified
Variable terms: 2a, 5b^2; Constant term: 6

Step by step solution

01

- Identify the Terms

List all the terms in the given expression. The expression is \(2a + 5b^2 + 6\). The terms are separated by addition or subtraction. So, the terms are: \[ 2a, 5b^2, 6 \].
02

- Identify Variable Terms

A variable term contains a variable (like \(a\) or \(b\)). In the given expression, \(2a\) and \(5b^2\) are the variable terms. This is because they include the variables \(a\) and \(b\) respectively.
03

- Identify Constant Terms

A constant term is a term that doesn鈥檛 contain any variables, it's just a number. In the given expression, \(6\) is the constant term because it does not include any variables.

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Key Concepts

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

variable terms
Variable terms are key elements in algebra. They include any term that contains a variable, such as letters like \(a\), \(b\), or \(x\). These terms can represent varying quantities and are often combined with coefficients, which are the numerical parts attached to the variables.
In the expression \(2a + 5b^2 + 6\), the variable terms are \(2a\) and \(5b^2\):
  • In \(2a\), the variable is \(a\) and the coefficient is 2.
  • In \(5b^2\), the variable is \(b^2\) and the coefficient is 5.
These terms can change depending on the values substituted for the variables, making them dynamic components of algebraic expressions.
constant terms
Constant terms are quite different from variable terms. They are fixed values that do not change because they do not contain any variables. In the expression \(2a + 5b^2 + 6\), the constant term is \(6\).
Here's a closer look at the features of constant terms:
  • They are solely numbers without any attached variables.
  • They remain the same, regardless of the values substituted for any variables in the expression.
Constant terms provide a stable reference point within an algebraic expression, balancing the variable terms.
algebraic expressions
Understanding algebraic expressions is essential in prealgebra. These expressions are a combination of variable terms, constant terms, and operators (like addition or subtraction).
An example is \(2a + 5b^2 + 6\), which includes:
  • Variable terms: \(2a, 5b^2\)
  • Constant term: \(6\)
  • Operators: '+' connecting the terms
Algebraic expressions allow us to formulate and solve problems by representing real-life situations in a mathematical form. They lay the foundation for equations and inequalities in more advanced algebra.
prealgebra
Prealgebra introduces fundamental principles that prepare students for more complex algebra topics. It covers the basics such as understanding variable and constant terms in algebraic expressions.
Key areas of focus in prealgebra include:
  • Recognizing and differentiating between variable terms and constant terms.
  • Learning how to simplify and evaluate algebraic expressions.
  • Understanding properties of arithmetic applied to algebraic contexts.
By mastering these basic concepts, students build a strong foundation, easing their transition into higher-level algebra where they will encounter more intricate expressions and equations.

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Apply the appropriate property to simplify the expression. $$-3(25-m)$$

Demonstrate the commutative property of multiplication by evaluating the expressions for \(c=12\) and \(d=-4\). a. \(c \cdot d\) b. \(d \cdot c\)
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