Problem 19 - Technical Mathematics with Calculus

Question

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{x^{2}-4}{x^{3}-8}$$

Step-by-Step Solution

Answer

\(\frac{x + 2}{x^2+2x+4}\)

Step 1: Factor Numerator and Denominator

Factor the difference of squares in the numerator and the difference of cubes in the denominator. The numerator is a difference of squares: \(x^{2}-4 = (x + 2)(x - 2)\) and the denominator is a difference of cubes: \(x^{3}-8 = (x-2)(x^2+2x+4)\).

Step 2: Cancel Common Factors

Cancel the common \((x - 2)\) term from the numerator and the denominator.

Step 3: Write the Simplified Fraction

Write the simplified fraction without the cancelled term \((x - 2)\), resulting in \(\frac{x + 2}{x^2+2x+4}\).

Key Concepts

Difference of Squares

Understanding the difference of squares is crucial when simplifying algebraic expressions. It refers to an expression of the form \( a^2 - b^2 \), which can be factored into \( (a + b)(a - b) \). This is possible because when you expand \( (a + b)(a - b) \), you get \( a^2 - ab + ab - b^2 \), and the middle terms cancel each other out, leaving you with the original form, \( a^2 - b^2 \).

For example, if we have \( x^2 - 4 \), and recognize that 4 is a perfect square \( (2^2) \), we can apply the difference of squares rule to factor it as \( (x + 2)(x - 2) \). This simplification can make further steps in our problem-solving much easier, such as canceling out common factors that might appear in the numerator and the denominator of a fraction.

Difference of Cubes

The difference of cubes is another factoring technique similar to the difference of squares but is used when dealing with cubic terms. The general form is \( a^3 - b^3 \) which factors into \( (a - b)(a^2 + ab + b^2) \). This pattern arises from the fact that the product of these two factors results in the original cubic difference.

Take the denominator of our exercise, \( x^3 - 8 \). Recognizing that 8 is a perfect cube \( (2^3) \), we can factor it as \( (x - 2)(x^2 + 2x + 4) \). Remembering formulas like the difference of cubes can greatly aid in breaking down complex algebraic expressions into simpler components.

Canceling Common Factors

Canceling common factors in a fraction is a way to simplify it to its lowest terms. It involves dividing both the numerator and the denominator by the same nonzero term. Essentially, it's simplification by 'crossing out' shared factors.

In the provided example, after factoring, we observed that \( (x - 2) \) is present both in the numerator and the denominator. Because anything divided by itself is 1, we cancel the \( (x - 2) \) terms, which simplifies the fraction. One key piece of advice is to ensure you've fully factored both the numerator and the denominator before canceling; this step relies on the accuracy of your factoring.

Negative Exponents

When working with negative exponents, it's important to understand that they indicate the reciprocal of the base raised to the positive exponent. In simpler terms, \( a^{-n} = \frac{1}{a^n} \).

This knowledge helps in simplifying expressions so they don't contain any negative exponents. In the context of rational expressions, if, after simplifying, you're left with negative exponents, you'd rewrite them with positive exponents by moving them to the other part of the fraction. For example, if the numerator has a term with a negative exponent, you'd move it to the denominator with a positive exponent, and vice versa.