Problem 4 - Algebra and Trigonometry

Question

find all numbers that must be excluded from the domain of each rational expression. $$ \frac{x+7}{x^{2}-49} $$

Step-by-Step Solution

Answer

The numbers that need to be excluded from the domain of the given rational expression are 7 and -7.

Step 1: Identify the Denominator

The first step is to identify the denominator in the given expression, which is \(x^2-49\). This is what we need to zero in on.

Step 2: Factor the Denominator

The second step is to factor the denominator. Given that \(x^2-49\) is a difference of squares, it can be factored into \((x-7)(x+7)\).

Step 3: Find the Excluded Values

Lastly, we have to find the values that will make the expression in the denominator zero as these are the values need to be excluded from the domain. Setting each factor equal to zero, the solutions are found to be 7 and -7. Therefore, 7 and -7 are the numbers that must be excluded from the domain.

Key Concepts

Domain Exclusion

When dealing with rational expressions, it's important to determine which values cannot be included in the domain. This is because certain values can make the expression undefined. In mathematics, division by zero is not allowed and must be avoided.

To find the domain exclusions:

• Identify the denominator of the rational expression. For our example, this is \(x^2-49\).
• Set the denominator equal to zero and solve for \(x\). This helps find the values that make the expression undefined.
• For \(x^2-49=0\), solving gives \(x=7\) and \(x=-7\). These values cause the denominator to be zero, so they must be excluded from the domain.

The excluded values are \(x=7\) and \(x=-7\), since they lead to division by zero in the original expression \(\frac{x+7}{x^{2}-49}\).

Factorization

Factorization is a key skill in simplifying rational expressions. It involves breaking down a complex expression into simpler, multiply-together parts. Recognizing certain types of expressions can aid in easier factorization.

When faced with a quadratic like \(x^2 - 49\), we look to apply factoring techniques. The expression \(x^2 - 49\) is a classic case of the difference of squares, which simplifies quickly.

Steps for factorization:

• Recognize the form \(a^2 - b^2\).
• Apply the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\).
• For \(x^2 - 49\), rewrite as \((x-7)(x+7)\).

This technique breaks down an expression into factors, aiding in solving, simplifying, and identifying domain exclusions.

Difference of Squares

The difference of squares is a specific algebraic formula used to factor expressions. This occurs when you have two square numbers separated by a subtraction sign. For example, \(x^2 - 49\) fits this pattern.

The general formula is given by:

• \(a^2 - b^2 = (a - b)(a + b)\)
• For \(x^2 - 49\), identify \(a\) as \(x\) and \(b^2\) as \(7^2\).
• The factorization is \((x-7)(x+7)\), where \(7\) and \(-7\) become the roots for exclusions.

Applying the formula makes it easy to identify excluded values and simplify expressions. The factorization process leverages this formula to quickly resolve quadratic expressions into their component parts.