Problem 18 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity.. The product \(P\) of an integer \(n\) and twice the integer

Step-by-Step Solution

Answer

The formula is \( P = 2n^2 \).

Step 1: Understand the Problem

We need to write an algebraic formula that represents the product of an integer \(n\) and twice the integer. This means we are dealing with two values: \(n\) and \(2n\). Our task is to find the product of these two values.

Step 2: Define the Quantities

Identify and define the two quantities mentioned in the problem. The first quantity is the integer itself, represented as \(n\). The second quantity is twice the integer, which can be expressed as \(2n\).

Step 3: Write the Product Formula

The word 'product' suggests multiplication. Therefore, we need to multiply the two defined quantities. The algebraic formula representing this product is: \[ P = n \times 2n \]

Step 4: Simplify the Formula

To simplify the formula, multiply \(n\) by \(2n\) to get: \[ P = 2n^2 \] This is the simplified algebraic expression for the product of an integer and twice the integer.

Key Concepts

Integer Multiplication

Integer multiplication is a basic arithmetic operation where two integers are multiplied together to find their product. When dealing with integers, remember that:

• Multiplying a positive integer by another positive integer gives a positive result.
• Multiplying a positive integer by a negative integer gives a negative result.
• Multiplying two negative integers gives a positive result.

In our exercise, we are focusing on two integers: one is our variable \( n \), and the other is \( 2n \), which is described as twice the integer. Integer multiplication helps us transform these terms into a product that can be easily simplified through mathematical operations. Knowing these basics ensures that multiplication with positive or negative integers leads to accurate results.

Algebraic Formula

An algebraic formula is a way of expressing a mathematical relationship using symbols and numbers. It's like a rule that describes how different quantities are related.
In the given exercise's context, you are asked to write an algebraic formula that represents the relationship between an integer \( n \) and twice that integer, which means multiplying them together. Thus the formula is: \[ P = n \times 2n \]
Using algebra, we can replace verbal descriptions with mathematical symbols to simplify communication and calculation. Algebraic formulas are essential since they provide a uniform way to describe relationships, making problem-solving more straightforward. Once you establish and understand the formula, like our \( P = n \times 2n \), you can apply rules and operations, such as simplification, to make these formulas more manageable.

Product of Integers

The product of integers refers to the result obtained from multiplying two or more integers. It is a straightforward concept, though, mastering it is crucial for more complex mathematics.
In our problem, we initially took the integers \( n \) and \( 2n \) to determine their product. By multiplying them, we apply the principle that multiplication combines numbers to form a larger number, referred to as the product.
In our specific task, this multiplication results in: \[ P = n \times 2n = 2n^2 \]
The simplified form \( 2n^2 \) demonstrates how multiplying integers results in products that can be transformed further to reflect more practical or simplified expressions. Understanding the product of integers is vital because it sets the stage for working with more sophisticated algebraic expressions, allowing for deeper exploration of mathematical relationships.