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Chapter 4: Problem 19

For Problems \(9-50\), simplify each rational expression. $$ \frac{x^{2}-4}{x^{2}+2 x} $$

Short Answer

Expert verified
The simplified expression is \(\frac{x-2}{x}\).

Step by step solution

01

Factor the Numerator

The numerator of the rational expression is \(x^{2}-4\). This is a difference of squares and can be factored as \((x-2)(x+2)\).
02

Factor the Denominator

The denominator is \(x^{2}+2x\). Factor out the greatest common factor, which is \(x\), to get \(x(x+2)\).
03

Simplify by Canceling Common Factors

Now that both the numerator, \((x-2)(x+2)\), and the denominator, \(x(x+2)\), are factored, we can cancel the common factor \((x+2)\) from the numerator and the denominator. This leaves us with \(\frac{x-2}{x}\).
04

State the Simplified Expression

After cancelling the common factors, the simplified rational expression is \(\frac{x-2}{x}\).

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Key Concepts

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

Factoring Polynomials
When dealing with expressions containing polynomials, factoring is a crucial skill. It involves breaking down a polynomial into simpler, multiplied expressions called factors. Factoring helps in simplifying complex expressions and solving polynomial equations.
  • Identify any common factors in all terms. For example, in the expression \( x^2 + 2x \), both terms share a common factor of \( x \).
  • Consider special factoring formulas like the difference of squares for expressions of the form \( a^2 - b^2 \), which factors into \((a-b)(a+b)\).
In our exercise, the numerator \( x^2 - 4 \) is a perfect example of the difference of squares, resulting in \((x-2)(x+2)\). The denominator \( x^2 + 2x \) can be factored by extracting \( x \) as the common factor, giving us \( x(x+2) \). These steps make it easier to simplify later.
Difference of Squares
The difference of squares is a specific type of polynomial that can be factored quickly when recognized. It follows the pattern \( a^2 - b^2 = (a-b)(a+b) \). This pattern can be handy in simplifying expressions and solving equations.

For example, in the expression \( x^2 - 4 \), we see that it's written as \( x^2 - 2^2 \). Here, \( a \) is \( x \) and \( b \) is \( 2 \), fitting perfectly into the difference of squares formula.
  • Factor \( x^2 - 4 \) as \( (x-2)(x+2) \).
  • This factorization breaks it down into simpler terms that can easily be canceled in rational expressions.
Recognizing the difference of squares can save time and simplify solving steps, as shown in the solution to our problem.
Simplifying Fractions
Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.

In rational expressions, this involves canceling common factors in the numerator and the denominator to make the expression easier to handle.
  • After factoring both parts of the rational expression, such as \( \frac{(x-2)(x+2)}{x(x+2)} \), look for identical factors in the numerator and denominator.
  • In our example problem, \( (x+2) \) is a common factor in both.
  • Cancel this common factor to simplify the fraction to \( \frac{x-2}{x} \).
This step reflects a fundamental algebraic property: you can cancel terms only if they are multiplicative factors in both parts of the fraction. Simplifying helps us work with expressions more effectively and understand the underlying relationships between variables.

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Most popular questions from this chapter

For Problems 45-60, set up an algebraic equation and solve each problem. A sum of \(\$ 1750\) is to be divided between two people in the ratio of 3 to 4 . How much does each person receive?

For Problems \(31-44\), solve each equation for the indicated variable. $$ y=\frac{5}{6} x+\frac{2}{9} \text { for } x $$

For Problems \(1-44\), solve each equation. $$ \frac{-3}{4 x+5}=\frac{2}{5 x-7} $$

For Problems \(1-30\), solve each equation. $$ \frac{a}{a+2}+\frac{3}{a+4}=\frac{14}{a^{2}+6 a+8} $$

For Problems \(1-30\), solve each equation. $$ \frac{3}{n-5}+\frac{4}{n+7}=\frac{2 n+11}{n^{2}+2 n-35} $$
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