Problem 50 - Algebra for College Students

Question

Simplify. $$\frac{x-x^{-2}}{1-x^{-2}}$$

Step-by-Step Solution

Answer

\( \frac{x^2 + x + 1}{x+1} \)

Step 1: Rewrite using common denominator

Rewrite both the numerator and the denominator by expressing each term with a common denominator. Note that the common denominator for each term involving powers of x is \( x^2 \).The numerator becomes \( x - x^{-2} = x - \frac{1}{x^2} \).The denominator becomes \( 1 - x^{-2} = 1 - \frac{1}{x^2} \).

Step 2: Combine fractions

Combine the terms in the numerator and the denominator into a single fraction each.The numerator is \( x - \frac{1}{x^2} = \frac{x^3 - 1}{x^2} \).The denominator is \( 1 - \frac{1}{x^2} = \frac{x^2 - 1}{x^2} \).

Step 3: Simplify the complex fraction

Write the original expression as a division of two fractions, then simplify:\[ \frac{\frac{x^3 - 1}{x^2}}{\frac{x^2 - 1}{x^2}} = \frac{x^3 - 1}{x^2 - 1} \].

Step 4: Factor and simplify

Factor the numerator and the denominator:The numerator \( x^3 - 1 \) can be factored as \( (x-1)(x^2 + x + 1) \).The denominator \( x^2 - 1 \) can be factored as \( (x-1)(x+1) \).So, \( \frac{(x-1)(x^2 + x + 1)}{(x-1)(x+1)} \).

Step 5: Cancel common factors

Cancel the common factor \( (x-1) \) from the numerator and the denominator:\[ \frac{(x-1)(x^2 + x + 1)}{(x-1)(x+1)} = \frac{x^2 + x + 1}{x+1} \].

Key Concepts

simplification

Simplification involves making mathematical expressions easier to work with by combining like terms and reducing fractions. The aim is to create a more straightforward form that is easier to interpret and solve. In the given exercise, we simplify the complex fraction by combining fractions in both the numerator and the denominator.
To further break it down:

• We rewrite the numerator and denominator with common denominators to combine them into single fractions.
• Once combined, we express the original expression as a division of two fractions.
• Finally, we simplify the resulting fraction by factoring and cancelling common factors.

The goal here is to make the fraction as simple as possible by breaking down each portion into its simplest form.

factoring polynomials

Factoring polynomials is a key step in simplifying complex fractions. It involves breaking down a polynomial into products of simpler polynomials. This makes it easier to identify common factors that can be cancelled out.
For example:

• In the solution, the polynomial in the numerator is \( x^3 - 1 \).
• We factor this using the difference of cubes, giving us \( (x-1)(x^2 + x + 1) \).
• The polynomial in the denominator is \( x^2 - 1 \), which can be factored as the difference of squares, giving us \( (x-1)(x+1) \).

By factoring these polynomials, we can cancel the common factor \( (x-1) \), simplifying the fraction further. This step is crucial for reducing complex expressions into more manageable forms.

complex fractions

Complex fractions have fractions in either the numerator, the denominator, or both. Simplifying these usually involves a few systematic steps.
In the given exercise, we start by rewriting the terms in the numerator and denominator with a common denominator to combine them:

• Numerator: \( x - \frac{1}{x^2} \) becomes \( \frac{x^3 - 1}{x^2} \).
• Denominator: \( 1 - \frac{1}{x^2} \) becomes \( \frac{x^2 - 1}{x^2} \).

This converts the complex fraction into a division of two simpler fractions:

• \[ \frac{\frac{x^3 - 1}{x^2}}{\frac{x^2 - 1}{x^2}} = \frac{x^3 - 1}{x^2 - 1} \]

Now, we factor and simplify further:

• Factor the numerator and denominator.
• Cancel the common factors.

Understanding and practicing these steps will make handling complex fractions less daunting and more intuitive.