Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 43
Perform the operations. Simplify, if possible. $$ \frac{2}{a^{2}+4 a+3}+\frac{1}{a+3} $$
Short Answer
Expert verified
The simplified form is \(\frac{1}{a+1}\).
Step by step solution
01
Factor the Quadratic Expression
The first step is to factor the quadratic expression in the first denominator, \(a^2 + 4a + 3\). We look for two numbers that multiply to 3 (the constant term) and add up to 4 (the coefficient of the linear term). The numbers 1 and 3 satisfy these conditions. Thus, the expression factors to \((a+1)(a+3)\).
02
Simplify the Expression
With the quadratic expression factored, the original expression is rewritten as follows:\[\frac{2}{(a+1)(a+3)} + \frac{1}{a+3}\]To perform addition, we need a common denominator. The common denominator is \((a+1)(a+3)\).
03
Rewrite Second Fraction with Common Denominator
To rewrite \(\frac{1}{a+3}\) with the common denominator \((a+1)(a+3)\), multiply the numerator and the denominator by \(a+1\):\[\frac{1}{a+3} = \frac{1 \cdot (a+1)}{(a+3) \cdot (a+1)} = \frac{a+1}{(a+1)(a+3)}\]
04
Add the Fractions
Add the fractions now that they have a common denominator:\[\frac{2}{(a+1)(a+3)} + \frac{a+1}{(a+1)(a+3)} = \frac{2 + (a+1)}{(a+1)(a+3)}\]Combine the numerators:\[\frac{2 + a + 1}{(a+1)(a+3)} = \frac{a + 3}{(a+1)(a+3)}\]
05
Simplify if Possible
The expression \(\frac{a + 3}{(a+1)(a+3)}\) can be simplified by canceling the common term \((a+3)\) in the numerator and the denominator, yielding:\[\frac{1}{a+1}\]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Factoring
Polynomial factoring is a valuable tool in simplifying rational expressions. When you encounter a quadratic expression like \(a^2 + 4a + 3\), you can break it down into a product of simpler binomial factors.
The key to factoring is identifying two numbers that multiply to the constant term and add to the linear coefficient. In this example, the numbers 1 and 3 multiply to 3 and add up to 4. By factoring, you rewrite the expression as \((a+1)(a+3)\).
The key to factoring is identifying two numbers that multiply to the constant term and add to the linear coefficient. In this example, the numbers 1 and 3 multiply to 3 and add up to 4. By factoring, you rewrite the expression as \((a+1)(a+3)\).
- This step simplifies future operations, like finding a common denominator.
- Factoring is especially important because it reveals zeros of the polynomial.
Common Denominator
Finding a common denominator is crucial when adding fractions. For the expression \(\frac{2}{(a+1)(a+3)} + \frac{1}{a+3}\), both fractions must share the same denominator to be combined.
The least common denominator (LCD) is the smallest expression that both denominators can divide without leaving a remainder. In this case, the LCD is \((a+1)(a+3)\).
The least common denominator (LCD) is the smallest expression that both denominators can divide without leaving a remainder. In this case, the LCD is \((a+1)(a+3)\).
- Using the LCD ensures the fractions have a common base for addition.
- This step is essential when merging or simplifying rational expressions.
Fraction Addition
Once both fractions have a common denominator, the process of addition becomes straightforward. You simply combine the numerators over the shared denominator. For example:
\[ \frac{2}{(a+1)(a+3)} + \frac{a+1}{(a+1)(a+3)} = \frac{2 + (a+1)}{(a+1)(a+3)} \]Simplify the numerator by combining like terms to arrive at:
\[ \frac{a + 3}{(a+1)(a+3)} \]
\[ \frac{2}{(a+1)(a+3)} + \frac{a+1}{(a+1)(a+3)} = \frac{2 + (a+1)}{(a+1)(a+3)} \]Simplify the numerator by combining like terms to arrive at:
\[ \frac{a + 3}{(a+1)(a+3)} \]
- Notice how the numerator is now one single expression.
- The common denominator remains unchanged.
Expression Simplification
The final step in dealing with rational expressions is simplification. With the simplified fraction \(\frac{a+3}{(a+1)(a+3)}\), check if you can cancel any terms.
Here, \(a+3\) appears in both the numerator and denominator, allowing you to cancel it, which yields:\[ \frac{1}{a+1} \]
Here, \(a+3\) appears in both the numerator and denominator, allowing you to cancel it, which yields:\[ \frac{1}{a+1} \]
- Simplification reduces the expression to its simplest form, making it easier to interpret and use in further calculations.
- Remember that cancellation is only possible when the same factor appears in both the numerator and the denominator.
