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Chapter 6: Problem 50

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

Identify the Rational Expression

The given rational expression is \( \frac{r^3 - s^3}{r - s} \). To simplify, we need to factor the numerator or the denominator.
02

Factor the Numerator

Recognize that the numerator \( r^3 - s^3 \) is a difference of cubes. The difference of cubes can be factored using the formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). So, \( r^3 - s^3 \) becomes \( (r - s)(r^2 + rs + s^2) \).
03

Substitute the Factored Form

Now substitute the factored form of the numerator back into the rational expression: \( \frac{(r - s)(r^2 + rs + s^2)}{r - s} \).
04

Cancel Common Factors

Cancel the common factor \( r - s \) in the numerator and the denominator. This simplifies to \( r^2 + rs + s^2 \).
05

Write the Simplified Expression

The expression in the lowest terms is \( r^2 + rs + s^2 \).

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Key Concepts

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

difference of cubes
To understand the simplification of the rational expression, it's important to grasp the concept of the difference of cubes. The difference of cubes is a special algebraic identity that helps in factoring expressions of the form \(a^3 - b^3\). The formula for factoring a difference of cubes is: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\].
In our exercise with \(r^3 - s^3\), we identify \(a = r\) and \(b = s\).
This allows us to rewrite \(r^3 - s^3\) using the difference of cubes formula: \(r^3 - s^3 = (r - s)(r^2 + rs + s^2)\).
Recognizing and applying this special factorization is a crucial first step in simplifying the rational expression.
factoring
Factoring is the process of breaking down an expression into products that can multiply together to give the original expression. In our case, we used the difference of cubes formula to factor the numerator \( r^3 - s^3 \) into \( (r - s)(r^2 + rs + s^2) \).
By factoring, we make the rational expression more manageable. Instead of dealing with a cubic expression, we now have a product of a binomial and a trinomial.
Factoring helps reveal common factors in the numerator and denominator of rational expressions. This is beneficial when simplifying, as common factors can be canceled out to simplify the expression further.
canceling common factors
Canceling common factors is essential in simplifying rational expressions. After factoring the numerator \( r^3 - s^3 \) as \( (r - s)(r^2 + rs + s^2)\), we notice that the binomial \( r - s \) appears in both the numerator and denominator.
By canceling the common factor \( r - s \) from both the numerator and the denominator, we are left with \( r^2 + rs + s^2 \).
This leaves us with the simplified form of the rational expression.
This step is crucial, as it often transforms complex fractions into simpler forms that are easier to understand and work with.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{a^{3}-8 b^{3}}{a^{2}-a b-6 b^{2}} \cdot \frac{a^{2}+a b-12 b^{2}}{a^{2}+2 a b-8 b^{2}} $$

Which rational expressions are equivalent to \(-\frac{x}{y} ?\) A. \(\frac{-x}{-y}\) B. \(\frac{x}{-y}\) C. \(\frac{x}{y}\) D. \(-\frac{x}{-y}\) E. \(\frac{-x}{y}\) F. \(-\frac{-x}{-y}\)

Give the domain of each rational function using (a) set-builder notation and (b) interval notation. $$ f(x)=\frac{9 x^{2}-8 x+3}{4 x^{2}+1} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{3}{8} \div \frac{9}{14} $$

Solve each problem. The volume of gas varies inversely as the pressure and directly as the temperature. (Temperature must be measured in kelvins (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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