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Chapter 7: Problem 46

Write each rational expression in lowest terms. $$ \frac{r^{3}-s^{3}}{r-s} $$

Short Answer

Expert verified
\(r^2 + rs + s^2\)

Step by step solution

01

Understand the Problem

The goal is to simplify the rational expression \(\frac{r^3 - s^3}{r - s}\) by writing it in its lowest terms.
02

Factor the Numerator

Recognize that \(r^3 - s^3\) is a difference of cubes. The formula for factoring a difference of cubes is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(r\) corresponds to \(a\) and \(s\) corresponds to \(b\). Therefore, factor the numerator as follows: \(r^3 - s^3 = (r - s)(r^2 + rs + s^2)\).
03

Simplify the Expression

The rational expression now looks like this: \(\frac{(r-s)(r^2 + rs + s^2)}{r - s}\). Cancel the common factor \(r - s\) in the numerator and denominator. This leaves us with \(r^2 + rs + s^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring polynomials
Polynomials are mathematical expressions that consist of variables and coefficients. They are often combined through addition, subtraction, and multiplication. Factoring polynomials is the process of breaking down these complex expressions into simpler 'factors' that, when multiplied together, give the original polynomial.

Key steps to factoring:
  • Identify the greatest common factor (GCF) of all terms in the polynomial.
  • Look for special factoring formulas, like the difference of squares, the sum or difference of cubes, or perfect square trinomials.
  • If there are no special formulas, apply factoring techniques like grouping or using the quadratic formula.

For instance, in the given exercise, the numerator \(r^3 - s^3\) is a type of polynomial known as the difference of cubes. By recognizing this, we can apply a specialized formula to simplify the expression before proceeding further.
difference of cubes
The difference of cubes is a special polynomial expression of the form \(a^3 - b^3\). This kind of polynomial can be factored using a specific formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

In our exercise, \(r^3 - s^3\) fits this structure, where \(r\) corresponds to \(a\) and \(s\) corresponds to \(b\). Therefore, we can factor it as follows: \(r^3 - s^3 = (r - s)(r^2 + rs + s^2)\).

Using this formula simplifies the numerator and prepares it for the next step in simplification. Remember, recognizing these patterns is crucial for simplifying more complex polynomial expressions efficiently.
canceling common factors
After factoring the polynomial, the next step is to simplify the rational expression by canceling common factors that appear in both the numerator and the denominator.

Rational expressions are similar to fractions in that you can reduce them by dividing out common factors. In the context of our exercise, the expression \frac{r^3 - s^3}{r - s}\ can be simplified once the numerator is factored: \frac{(r-s)(r^2 + rs + s^2)}{r - s}\.

Steps to cancel common factors:
  • Factor both the numerator and denominator.
  • Identify any common factors between the two.
  • Divide both the numerator and the denominator by the common factor.

Here, \(r - s\) appears in both the numerator and the denominator, so we can cancel it out, leaving us with \(r^2 + rs + s^2\) which is the simplified form of the rational expression.

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Most popular questions from this chapter

Solve each problem mentally. Use proportions in Exercises 23 and 24. In a certain southern state, sales tax on a purchase of \(\$ 1.50\) is \(\$ 0.12 .\) What is the sales tax on a purchase of \(\mathrm{S} 9.00 ?\)

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5 x-y}{x^{2}+x y-2 y^{2}}-\frac{3 x+2 y}{x^{2}+5 x y-6 y^{2}} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{6 x+5 y}{6 x^{2}+5 x y-4 y^{2}}-\frac{x+2 y}{9 x^{2}-16 y^{2}} $$

Write each rational expression in lowest terms. $$ \frac{5 k-10}{20-10 k} $$

Solve each problem. See Examples \(1-7\) In \(2007,51.2 \%\) of the homes in the United States used natural gas as the primary heating fuel. (Source: U.S. Census Bureau.) The volume of gas varies inversely as the pressure and directly as the temperature. (Temperature must be measured in Kelvin (K), a unit of measurement used in physics.) If a certain gas occupies a volume of \(1.3 \mathrm{L}\) at \(300 \mathrm{K}\) and a pressure of 18 newtons, find the volume at \(340 \mathrm{K}\) and a pressure of 24 newtons.
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