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Chapter 7: Problem 21

For exercises 1-66, simplify. $$ \frac{5}{5 x^{2}+10 x} $$

Short Answer

Expert verified
\(\frac{1}{x(x + 2)}\)

Step by step solution

01

Factor the Denominator

Factor the common term out of the denominator. The expression in the denominator is: \[5x^2 + 10x\]. We can factor out a 5x: \[5x(x + 2)\]
02

Rewrite the Expression

Rewrite the original fraction with the factored denominator: \[\frac{5}{5x(x + 2)}\]
03

Simplify the Fraction

Divide out the common terms in the numerator and denominator. The common term is 5: \[\frac{1}{x(x + 2)}\]

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Key Concepts

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

Factoring Polynomials
To simplify algebraic fractions, it's important to first understand factoring polynomials. Factoring involves breaking down a polynomial into simpler 'factor' components that, when multiplied together, give the original polynomial.
In our exercise, we start with the denominator: \[5x^2 + 10x\]. We need to find common factors in these terms to factor it.
Notice, both terms share a factor of 5x. Therefore, we factor out 5x:
\[5x^2 + 10x = 5x(x + 2)\]
Now, our denominator is in its simplest factored form, making it easier to simplify the fraction.
Common Factors
Identifying and removing common factors is key in simplifying algebraic fractions. Common factors are numbers or expressions that divide exactly into all the terms of a polynomial or fraction.
In the original exercise, we have \[ \frac{5}{5x(x+2)} \].
Look at both the numerator and the factored denominator, we see that both include the common factor of 5. We can divide both the numerator and the denominator by this common factor to simplify further.
This process of canceling out the common factors is a crucial step for simplifying fractions:
\[ \frac{5}{5x(x+2)} = \frac{1}{x(x+2)} \]
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying these expressions often involves factoring and canceling out common terms.
In our exercise, we worked with the rational expression \[ \frac{5}{5x^2 + 10x} \].
By factoring the denominator and canceling common factors, we derived the simplified form \[ \frac{1}{x(x+2)} \].
This process reduces a complex fraction to its simplest form, making it easier to work with. Understanding how to manipulate and simplify rational expressions is essential for performing various algebraic operations.

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Most popular questions from this chapter

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{3} x+\frac{3}{2}=\frac{1}{3} x+\frac{1}{6} $$

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