Problem 91 - Algebra for College Students

Question

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{1}{x-1}=\frac{?}{x^{3}-1}$$

Step-by-Step Solution

Answer

? = x^2 + x + 1

Step 1: Identify the given equation

The given equation is \(\frac{1}{x-1}=\frac{?}{x^3-1}\).

Step 2: Factorize the denominator on the right side

Factorize \(x^3-1\) using the difference of cubes formula: \(x^3-1=(x-1)(x^2+x+1)\). The equation now looks like \(\frac{1}{x-1}=\frac{?}{(x-1)(x^2+x+1)}\).

Step 3: Match the denominators

To make the denominators match, the expression on the right side must be multiplied by \(x^2+x+1\) in the numerator since \(x^3-1=(x-1)(x^2+x+1)\).

Step 4: Write the equivalent expression

Multiply the numerator \(1\) by \(x^2+x+1\): \( \frac{1 \times (x^2+x+1)}{(x-1)(x^2+x+1)}= \frac{x^2+x+1}{x^3-1} \). Therefore, \(? = x^2 + x + 1\).

Key Concepts

factoring polynomials

Factoring polynomials is like breaking them down into simpler pieces that, when multiplied together, give you the original polynomial. It's similar to factoring numbers, like how 6 can be factored into 2 and 3. For polynomials, you might be looking at terms like \(ax^2 + bx + c\). Common factoring techniques include finding common factors, grouping, and special formulas like the difference of squares or cubes. In our problem, we specifically used the difference of cubes formula to factor \(x^3 - 1\). Understanding how to factor helps simplify expressions and solve equations more easily.

difference of cubes

The difference of cubes is a special polynomial form: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]. It allows us to break down a cubic expression into factors. In our example, \(x^3 - 1\) is \(x^3 - 1^3\). Using the formula, we get \(x^3 - 1 = (x - 1)(x^2 + x + 1)\). This is essential for simplifying the given rational expression. Once factored, it becomes easier to match denominators and find equivalent expressions.

equivalent fractions

Equivalent fractions represent the same value, even if they look different. For rational expressions (fractions containing polynomials), you need to have the same denominator to compare or combine them. For example, 1/2 and 2/4 are equivalent because they reduce to the same value. In our problem, we set the denominator of the simplified fraction equal to \(x^3 - 1\) by including \(x^2 + x + 1\) in the numerator. This way, both sides of the equation represent the same rational expression, just in different forms. Understanding equivalency ensures you're working with expressions correctly and consistently.