Problem 14 - Algebra and Trigonometry

Question

simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{x^{2}-14 x+49}{x^{2}-49} $$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{x-7}{x+7}\) and the domain is all real numbers except -7.

Step 1: Factorize the expression

Begin by factorizing the expression. The numerator can be factored into \((x-7)^2\) and the denominator can be factored into \((x-7)(x+7)\). So the expression becomes \(\frac{(x-7)^2}{(x-7)(x+7)}\)

Step 2: Simplify the expression

Next, simplify the expression by cancelling out the common factors in the numerator and the denominator. Here, one \((x-7)\) from the numerator can get cancelled with the \((x-7)\) in the denominator, obtaining \(\frac{x-7}{x+7}\).

Step 3: Find the domain

The last step is to find the domain. The domain is all real numbers except for those that make the denominator equal to 0. Solve the equation \(x+7 = 0\) to find the values that need to be excluded. Solving for x gives \(x = -7\). Therefore, all real numbers except -7 are included in the domain.

Key Concepts

Domain Exclusion

When working with rational expressions, it's crucial to identify the domain exclusion. The domain of a function is the set of all possible input values (x-values). However, for rational expressions, there can be values that make the denominator zero, which leads to an undefined expression.

To find these exclusions, set the denominator of the rational expression equal to zero and solve for the variable. For the given problem, we have the expression \( \frac{x^2 - 14x + 49}{x^2 - 49} \). The denominator \( x^2 - 49 \) must not be zero. Setting \( x^2 - 49 = 0 \) and solving it, you get:

• \( x^2 = 49 \)
• \( x = 7 \) or \( x = -7 \)

Thus, \( x = 7 \) and \( x = -7 \) are domain exclusions because substituting these into the denominator gives zero. Keeping these values out ensures the rational expression remains valid.

Factoring Polynomials

Factoring is an essential skill in simplifying expressions and solving equations. When you factor a polynomial, you break it down into simpler "factor" polynomials. For our problem, we factor both the numerator and the denominator.

The numerator \( x^2 - 14x + 49 \) can be factored as \((x-7)^2\). This is because the expression is a perfect square trinomial.

• A perfect square trinomial fits the pattern \((a-b)^2 = a^2 - 2ab + b^2\).
• In this case, \( a = x \) and \( b = 7 \), giving us \((x-7)^2 \).

Similarly, the denominator \( x^2 - 49 \) is a difference of squares and factors into \((x-7)(x+7)\). The pattern here is \( a^2 - b^2 = (a-b)(a+b) \).

Factoring simplifies the problem and reveals opportunities to simplify the expression further by canceling common factors.

Simplifying Expressions

Simplification is the process of reducing an expression to its simplest form. After factoring, you look for common factors in the numerator and denominator that can be cancelled.

In the expression \( \frac{(x-7)^2}{(x-7)(x+7)} \), we notice that there's a common factor \((x-7)\). Cancel one instance of \( (x-7) \) from the numerator and the denominator, which simplifies the expression to \( \frac{x-7}{x+7}\).

It's important to note that canceling doesn’t change the domain exclusions identified previously; the original domain restrictions still apply to the simplified expression because the simplification process doesn’t eliminate undefined points in the domain. The x-values \( x = 7 \) or \( x = -7 \) remain exclusions.

• Always check the domain again after simplifying to ensure it reflects any canceled terms.

After simplifying, you often have a more manageable expression and a report on its domain.