Problem 36 - Elementary Algebra Within Reach

Question

In Exercises 23-36, write a verbal description of the algebraic expression, without using a variable. (There is more than one correct answer.) $$ x^{3}-1 $$

Step-by-Step Solution

Answer

A verbal description of the expression \(x^{3}-1\) could be: 'One less than a certain number cubed' or 'A number cubed, then reduced by one'.

Step 1: Understand the Algebraic Expression

The expression \(x^{3}-1\) can be broken down into two parts: \(x^{3}\) and \(-1\). \(x^{3}\) can be described as a number or quantity that was cubed, and \(-1\) describes the subtraction of one unit from that cubed number or quantity.

Step 2: Create Verbal Description

Each part of the expression now has a verbal description: 'A number cubed' for \(x^{3}\) and 'minus one' for \(-1\). Combine these into a complete sentence reflecting the operation in the entire expression.

Step 3: Verbal Description Complete

Putting the two parts together, a possible verbal description of \(x^{3}-1\) could be: 'One less than a certain number cubed' or 'A number cubed, then reduced by one'.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra, used to represent numbers, operations, and variables in a concise way. Instead of dealing with specific numbers, algebraic expressions allow us to work with values in general terms which can be manipulated, evaluated, and simplified.
Common components of an algebraic expression include:

• Constants: Specific numbers like 1, 2, or 3.
• Variables: Symbols, often represented by letters such as x, y, or z, which stand in for unknown or variable quantities.
• Operators: Mathematical symbols like plus (+), minus (-), multiplication (×), and division (÷) which describe the operations to be performed.

In our example, the expression is \(x^{3}-1\), where \(x\) is a variable, \(x^{3}\) is a term indicating the cubing of this variable, and "-1" signifies the subtraction of one unit.
This expression can be verbally described as 'One less than a cube of a number.' Such descriptions are useful when conveying the meaning of expressions without direct numerical references.

Cubing in Algebra

Cubing in algebra refers to the operation of raising a number to the power of three. In mathematical notation, this is expressed as \(x^3\), indicating that the number \(x\) is multiplied by itself three times: \(x \times x \times x\).
Cubing increases the magnitude of a number significantly:

• Positive numbers: When cubed, they remain positive and grow larger.
• Negative numbers: A negative number cubed becomes more negative because a negative times a negative yields a positive, and a positive times another negative yields a negative.

In the context of our expression \(x^3 - 1\), the part \(x^3\) can be thought of as 'a number cubed', which is crucial for understanding the operation being described.
Cubing is a powerful transformation in algebra due to its ability to significantly alter the scale of numbers involved.

Subtraction in Algebra

Subtraction is a basic arithmetic operation represented by the minus sign (-), used to denote the removal or deduction of one quantity from another. In algebra, it's not only used with numbers but also with variables and expressions.
The subtraction sign directly impacts the terms it connects, changing the value of an expression. For instance, in our example \(x^3 - 1\), the \-1\ indicates a reduction of one unit from the cubed value of \(x\).
Key aspects of subtraction in algebra include:

• Order: Subtraction is not commutative; changing the order will change the result.
• Negative results: Subtracting a larger number from a smaller one results in a negative value.
• Expression simplification: Often used in simplifying expressions by combining like terms.

When crafting a verbal description, understanding subtraction's role helps clearly convey the intent of an expression, such as describing \(x^3 - 1\) as 'a number cubed, reduced by one'. This highlights the action of taking away or reducing the result after an operation.