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Chapter 6: Problem 49

Write each rational expression in lowest terms. $$ \frac{a^{3}+b^{3}}{a+b} $$

Short Answer

Expert verified
\(a^2 - ab + b^2\)

Step by step solution

01

Factor the numerator

Recognize that the numerator, \(a^3 + b^3\), is a sum of cubes. The sum of cubes can be factored using the formula: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Applying this to the given expression, the numerator becomes: \((a+b)(a^2 - ab + b^2)\).
02

Rewrite the expression

Substitute the factored form of the numerator into the original rational expression: \(\frac{a^3 + b^3}{a+b} = \frac{(a+b)(a^2 - ab + b^2)}{a+b}\).
03

Simplify the expression

Since \(a+b\) is a common factor in both the numerator and the denominator, they can be canceled out. This gives: \(\frac{a^2 - ab + b^2}{1}\).
04

Final simplified form

The rational expression in its lowest terms is: \(a^2 - ab + b^2\).

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Key Concepts

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

Factorization
Factorization is the process of breaking down an expression into simpler components, or factors, that when multiplied together, give the original expression. The main purpose is to simplify complex expressions.
In our example, we start with the numerator: \(a^3 + b^3\). This is recognized as a sum of cubes. The general formula for the sum of cubes is: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).
Using this formula, we factorize \(a^3 + b^3\) into: \( (a+b)(a^2 - ab + b^2)\). This step is crucial in simplifying the entire rational expression.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing them to their simplest form. A rational expression is a fraction where the numerator and the denominator are polynomials.
Our goal is to simplify \(\frac{a^3+b^3}{a+b}\). After factorizing the numerator as \( (a+b)(a^2 - ab + b^2)\), we rewrite the expression:
\(\frac{(a+b)(a^2 - ab + b^2)}{a+b}\)
In this form, we see \(a+b\) is a common factor in both the numerator and the denominator. These common factors can be canceled out. So, the simplified form of the expression is:
\(a^2 - ab + b^2\).
Always remember to look for common factors in both the numerator and the denominator to make simplification easier!
Sum of Cubes
The sum of cubes is a special algebraic expression and has a straightforward factorization formula. The sum of cubes formula is:
\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
Understanding this formula is essential because it helps in breaking down complex expressions into simpler ones, making them easier to handle and simplify.
For our given exercise, recognizing that \(a^3 + b^3\) is a sum of cubes and applying the formula lets us factorize the numerator efficiently.
Practice factorizing different sum of cubes expressions to become comfortable with this concept. It will help immensely in simplifying various rational expressions in algebra!

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{a c-a d+b c-b d}{a c-a d-b c+b d} $$

Add or subtract as indicated. $$\frac{2}{m^{2}-4 m+4}+\frac{3}{m^{2}+m-6}$$

Write each rational expression in lowest terms. $$ \frac{5 r-20}{3 r-12} $$

Add or subtract as indicated. $$\frac{3}{(p-2)^{2}}-\frac{5}{p-2}+4$$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{2}{3} \div \frac{8}{9} $$
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