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Chapter 0: Problem 103

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Short Answer

Expert verified
The result of the addition of the two rational expressions is \(\frac{10x+41}{(x+5)(x+2)}\).

Step by step solution

01

Identifying the Denominators

In the given expression, \(\frac{3}{x+5}+\frac{7}{x+2}\), the denominators are \(x+5\) and \(x+2\). These two expressions are the ones we will need to find a common denominator for.
02

Finding the Least Common Denominator

Since the denominators \(x+5\) and \(x+2\) have no common factors, the least common denominator will be the product of the two denominators: \((x+5)(x+2)\).
03

Rewriting the Fractions with the Common Denominator

Rewrite each fraction with the LCD as its denominator and adjust the numerator accordingly: \(\frac{3}{x+5} = \frac{3(x+2)}{(x+5)(x+2)}\) and \(\frac{7}{x+2} = \frac{7(x+5)}{(x+2)(x+5)}\).
04

Adding the Rational Expressions Together

Finally, we can add the two rational expressions together: \(\frac{3(x+2)+7(x+5)}{(x+5)(x+2)}\).
05

Simplifying the Result

Simplify the numerator and write the simplified result: \(\frac{3x+6+7x+35}{(x+5)(x+2)} = \frac{10x+41}{(x+5)(x+2)}\).

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