Problem 58 - Algebra and Trigonometry

Question

Factor the expression completely. $$ \left(1+\frac{1}{x}\right)^{2}-\left(1-\frac{1}{x}\right)^{2} $$

Step-by-Step Solution

Answer

The factored form of the expression is \( \frac{4}{x} \).

Step 1: Recognize the Difference of Squares

The expression \( \left(1 + \frac{1}{x}\right)^{2} - \left(1 - \frac{1}{x}\right)^{2} \) is in the form \( a^2 - b^2 \), where \( a = 1 + \frac{1}{x} \) and \( b = 1 - \frac{1}{x} \). This is a difference of squares, which can be factored into \( (a + b)(a - b) \).

Step 2: Substitute the Expressions

Substitute \( a = 1 + \frac{1}{x} \) and \( b = 1 - \frac{1}{x} \) into the factorization formula to get the expression: \[ \left( \left(1 + \frac{1}{x}\right) + \left(1 - \frac{1}{x}\right) \right) \left( \left(1 + \frac{1}{x}\right) - \left(1 - \frac{1}{x}\right) \right) \].

Step 3: Simplify the Terms

Simplify the first term: \( \left(1 + \frac{1}{x}\right) + \left(1 - \frac{1}{x}\right) = 2 \). Simplify the second term: \( \left(1 + \frac{1}{x}\right) - \left(1 - \frac{1}{x}\right) = \frac{2}{x} \).

Step 4: Multiply the Factors

Now multiply the simplified terms: \( 2 \times \frac{2}{x} = \frac{4}{x} \). The factored form of the expression is \( \frac{4}{x} \).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity that simplifies expressions of the form \( a^2 - b^2 \). It's called the "difference of squares" because it physically represents the subtraction (difference) of two perfect squares. The identity allows us to break it down into a product:

• \( (a + b)(a - b) \)

Here, \( a \) and \( b \) are any algebraic expressions. This transformation is handy because it reduces the degree of the expression, making it easier to handle.
For example, in the original problem, the expression is \( \left(1 + \frac{1}{x}\right)^2 - \left(1 - \frac{1}{x}\right)^2 \). Recognizing this as a difference of squares was critical to simplifying further.
Always look for opportunities to apply this identity when you see a polynomial subtraction.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This might involve factoring, combining like terms, or applying algebraic identities like the difference of squares. A simplified expression is usually easier to understand and work with in further calculations.
To simplify the expression in the exercise, we started by identifying it as a difference of squares. Then, after applying \( (a + b)(a - b) \), we simplified the inner terms:

• \( \left(1 + \frac{1}{x}\right) + \left(1 - \frac{1}{x}\right) = 2 \)
• \( \left(1 + \frac{1}{x}\right) - \left(1 - \frac{1}{x}\right) = \frac{2}{x} \)

Each step reduced the complexity of the expression, bringing us closer to the factored form \( \frac{4}{x} \). Remember, the goal is to make the expression as manageable as possible.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They are like shortcuts that simplify complex algebraic operations. The difference of squares, \( (a^2 - b^2) = (a+b)(a-b) \), is just one common example.
Using such identities effectively is crucial because they can turn a lengthy problem into a more straightforward task. They provide the groundwork for understanding deeper mathematics and are especially helpful in factoring, simplifying, and solving equations.
In this exercise, recognizing and applying the difference of squares was the key to factoring the expression quickly and efficiently. Students should familiarize themselves with various algebraic identities as they can greatly aid in simplifying complex problems.