Problem 2 - Algebra 2

Question

Write an algebraic expression to represent each verbal expression. twice a number decreased by the cube of the same number

Step-by-Step Solution

Answer

The expression is \( 2x - x^3 \).

Step 1: Identify Keywords

The phrase "twice a number" suggests that we are considering two times some unknown number, typically represented as a variable, say \( x \). The phrase "the cube of the same number" refers to this variable being raised to the power of three, or \( x^3 \). "Decreased by" indicates subtraction.

Step 2: Translate to Algebraic Expression

Based on the keywords identified, "twice a number" can be expressed as \( 2x \), and "the cube of the same number" as \( x^3 \). The complete verbal expression "twice a number decreased by the cube of the same number" translates to the algebraic expression \( 2x - x^3 \).

Key Concepts

Verbal Expressions

Verbal expressions are phrases that describe mathematical operations using words. These expressions are common in word problems and help us articulate mathematical ideas in our everyday language. Understanding verbal expressions is key to translating them into mathematical terms, a necessary skill in algebra.

One must be able to recognize common phrases and their mathematical equivalents to effectively work with verbal expressions:

• "Twice a number" means multiplying the number by two.
• Expressions like "sum of" or "difference between" might imply addition or subtraction respectively.
• "Times" can indicate multiplication, while "divided by" suggests division.

By breaking down a verbal expression, identifying keywords, and associating them with mathematical operations, it becomes easier to work with and solve problems that use this type of language.

Translation to Algebra

Translation to algebra involves converting verbal expressions into algebraic expressions by assigning symbols and operations. This translation process is crucial in solving real-world problems mathematically.

To translate effectively, follow these steps:

• Identify the mathematical operations indicated by the words. For example, "twice" translates to a multiplication by 2, and "decreased by" translates to a subtraction operation.
• Assign variables to unknown quantities. In the problem, "a number" can be represented by the variable \( x \).
• Construct the algebraic expression based on the identified keywords. For instance, "twice a number decreased by the cube of the same number" translates to \( 2x - x^3 \).

Each word in a verbal expression has a corresponding mathematical meaning, and understanding this relation is key to mastering algebraic translation.

Variables and Exponents

Variables and exponents are fundamental components of algebra that allow us to express and solve complex mathematical concepts.

Variables are symbols, typically letters, that represent unknown quantities or values. In our example, the variable \( x \) stands for "a number." Variables serve as placeholders that can have any value, allowing us to generalize problems and find solutions in algebra.

Exponents represent repeated multiplication of a number by itself. When we say "the cube of a number," it refers to raising the number to the third power, expressed as \( x^3 \). Here:

• The base is the variable \( x \).
• The exponent, \( 3 \), indicates how many times \( x \) is multiplied by itself.

Understanding variables and exponents is essential, as they help to simplify expressions and solve algebraic equations efficiently.