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Chapter 7: Problem 19

Add or subtract. Write answer in lowest terms. \(\frac{x^{2}}{x+5}+\frac{5 x}{x+5}\)

Short Answer

Expert verified
x

Step by step solution

01

Identify Common Denominator

Notice that both fractions have the same denominator, which is \(x+5\).
02

Combine the Numerators

Since the denominators are the same, the fractions can be added directly by combining the numerators:\[ \frac{x^{2} + 5x}{x+5} \]
03

Factor the Numerator (if possible)

Factor the numerator to see if it can be simplified further. In this case, there is a common factor of \x\ in the numerator:\[ \frac{x(x + 5)}{x + 5} \]
04

Simplify the Fraction

Since \(x + 5\) is in both the numerator and the denominator, they cancel each other out, leaving the simplified expression:\[ x \]

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Key Concepts

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

Common Denominator
When adding or subtracting rational expressions, the first step is to find a common denominator. This ensures that the fractions have the same base, making it easier to combine them.

In our exercise, both fractions already have the same denominator, which is great! The common denominator here is \({x+5}\).

Think of the common denominator as a shared foundation. If the denominators were different, we'd need to manipulate the fractions to make them the same by finding the Least Common Denominator (LCD).

But since we have the same denominator, we can move to the next step without extra work.
Factoring
Factoring is a critical part of simplifying expressions in algebra. After you've combined the fractions, watch out for opportunities to factor the numerator.

Factoring simplifies expressions and can sometimes reveal cancellations that reduce the fraction further.

In our exercise, once we combined the numerators, we got \(\frac{x^2 + 5x}{x + 5}\). Notice that the numerator, \(x^2 + 5x\), can be factored further by taking out the common factor x.

Factoring like this:
\(\frac{x(x + 5)}{x + 5}\)
reveals that both, numerator and denominator, have the term (x + 5). Excellent 鈥� now we can simplify even more!
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form, making them easier to understand and work with.

Once you've factored your expression, cancel common terms that appear in both, the numerator and the denominator.

In our exercise, the expression \( \frac{x(x + 5)}{x + 5}\) from the previous step can be simplified because (x + 5) is common to both, the numerator and the denominator.

By cancelling out (x + 5), you're essentially dividing it by itself, which equals 1. This leaves us with:
x
Congratulations, that's our simplified fraction!

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Most popular questions from this chapter

Solve each problem involving direct or inverse variation. If \(z\) varies directly as \(x,\) and \(z=30\) when \(x=8,\) find \(z\) when \(x=4\)

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