Problem 57 - Algebra for College Students

Question

Simplify. $$\frac{1-8 x^{-3}}{x^{-1}+2 x^{-2}+4 x^{-3}}$$

Step-by-Step Solution

Answer

\(x-2\)

Step 1: Simplify the Numerator

Rewrite the numerator by expressing each term with positive exponents if possible. The numerator is \(1 - 8x^{-3}\).

Step 2: Simplify the Denominator

Rewrite the denominator by expressing each term with positive exponents if possible. The denominator is \(x^{-1} + 2x^{-2} + 4x^{-3}\).

Step 3: Combine into a Single Fraction

Combine the expressions into a single fraction: \[ \frac{1 - 8x^{-3}}{x^{-1} + 2x^{-2} + 4x^{-3}} \]

Step 4: Factor Out the Smallest Exponent

Identify the smallest exponent of x in both numerator and denominator. The smallest in both is \(x^{-3}\). Factor \(x^{-3}\) out of the numerator and denominator: \[ \frac{x^{-3}(x^{3}-8)}{x^{-3}(x^{2} + 2x + 4)} \]

Step 5: Simplify the Fraction

Since \(x^{-3}\) is a common factor, cancel it out from both numerator and denominator: \[ \frac{x^{3} - 8}{x^{2} + 2x + 4} \]

Step 6: Recognize Special Formulas

\(x^{3} - 8\) is a difference of cubes, which can be factored further: \[ x^{3} - 8 = (x-2)(x^{2} + 2x + 4) \]

Step 7: Final Simplification

Substitute back the factored forms and cancel out the common terms: \[ \frac{(x-2)(x^{2} + 2x + 4)}{x^{2} + 2x + 4} = x-2 \]

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, if you see an expression like \(a^{-n}\), it means \(\frac{1}{a^n}\). This is especially handy when you need to simplify fractions that involve negative powers. In the given problem, you have \(x^{-3}\), \(x^{-2}\), and \(x^{-1}\). To convert these negative exponents into positive ones, you can rewrite them as follows:

• \(1 - 8x^{-3}\) becomes \(1 - \frac{8}{x^3}\)
• \(x^{-1} + 2x^{-2} + 4x^{-3}\) becomes \(\frac{1}{x} + \frac{2}{x^2} + \frac{4}{x^3}\)

Converting these terms into positive exponents facilitates easy factoring and simplification later on.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This step is crucial for simplifying rational expressions because it often reveals common factors that you can cancel out. In the given problem, after dealing with the exponents, we factor out the smallest power of x, which is \(x^{-3}\). This gives us: \( \frac{x^{-3}(x^3 - 8)}{x^{-3}(x^2 + 2x + 4)} \). Factoring out the smallest exponent simplifies the expression dramatically, allowing us to easily identify and cancel common factors in the next step.

Difference of Cubes

A difference of cubes is an algebraic identity given by \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Recognizing this form is essential in the simplification of certain expressions. In the provided problem, \(x^3 - 8\) can be identified as a difference of cubes: \(x^3 - 2^3\). You can factor this as \( (x - 2)(x^2 + 2x + 4) \). This reveals common terms in both numerator and denominator which can then be canceled out, making the expression far simpler.

Canceling Common Factors

Canceling common factors between the numerator and the denominator simplifies a rational expression. Once you have identified common terms, they can be removed, markedly simplifying the expression. In the provided exercise, we have: \( \frac{(x - 2)(x^2 + 2x + 4)}{x^2 + 2x + 4} \). Since \((x^2 + 2x + 4)\) appears in both the numerator and the denominator, they cancel out, leaving us with: \(x - 2\). This is the essence of simplifying rational expressions: reducing them to their simplest form.