Question

Simplify each expression. $$ \frac{7 x+35}{x^{2}+5 x} $$

   Simplify each expression.
$$
\frac{7 x+35}{x^{2}+5 x}
$$

Algebra A Combined Function

Algebra A Combined Function

Elayn Martin-Gay 4th Edition

Chapter 7, Problem 35

Instant Answer

Solved by Expert Victor Salazar

02/07/2022

Step 1

Step 1: First, we can factor out an $x$ from the denominator, which gives us: $$ \frac{7x+35}{x(x+5)} $$ Show more…

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Simplify each expression. $$ \frac{7 x+35}{x^{2}+5 x} $$

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Key Concepts

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. Understanding rational expressions is fundamental in algebra as it involves operations such as addition, subtraction, multiplication, division, and simplification. Recognizing how to work with these expressions is crucial for solving equations and understanding functions in a broader context.

Factoring

Factoring is the process of expressing a polynomial as a product of its simpler polynomial factors. It is a key algebraic technique used to simplify expressions, solve polynomial equations, and analyze the structure of expressions. In the context of simplifying rational expressions, factoring both the numerator and the denominator allows for the identification and cancellation of common factors.

Canceling Common Factors

Canceling common factors involves dividing both the numerator and denominator of a rational expression by their common polynomial factors. This technique is used to simplify the expression to its lowest terms. It is important to only cancel factors that are common to both the numerator and the denominator to ensure the expression remains equivalent to the original.

Domain Restrictions

When simplifying rational expressions, it is essential to consider domain restrictions, which are the values for the variable that make the original denominator equal to zero. These restrictions must be maintained even after canceling common factors, as they define where the expression is valid. Understanding domain restrictions ensures that the simplified expression accurately represents the behavior and limitations of the original expression.

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