In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (x²−14x+49)/(x²−49)

Verified step by step guidance

1. Factor both the numerator and the denominator of the rational expression. The numerator, \(x^2 - 14x + 49\), is a perfect square trinomial and can be factored as \((x - 7)^2\). The denominator, \(x^2 - 49\), is a difference of squares and can be factored as \((x - 7)(x + 7)\).
2. Write the rational expression in its factored form: \(\frac{(x - 7)^2}{(x - 7)(x + 7)}\).
3. Simplify the rational expression by canceling out one \((x - 7)\) term from the numerator and denominator. The simplified expression becomes \(\frac{x - 7}{x + 7}\).
4. Identify the values of \(x\) that must be excluded from the domain. These are the values that make the denominator equal to zero. Set \((x - 7)(x + 7) = 0\) and solve for \(x\). The excluded values are \(x = 7\) and \(x = -7\).
5. State the simplified rational expression and the domain restrictions. The simplified expression is \(\frac{x - 7}{x + 7}\), and the domain excludes \(x = 7\) and \(x = -7\).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions often involves factoring both the numerator and the denominator to identify common factors that can be canceled. Understanding how to manipulate these expressions is crucial for solving problems involving them.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For example, the expression x^2 - 14x + 49 can be factored into (x - 7)(x - 7) or (x - 7)^2. This process is essential for simplifying rational expressions and identifying restrictions on the variable.

Domain of a Rational Expression

The domain of a rational expression consists of all real numbers except those that make the denominator equal to zero. In the expression (x^2−14x+49)/(x^2−49), the denominator x^2 - 49 factors to (x - 7)(x + 7), indicating that x cannot be 7 or -7. Identifying these exclusions is vital for understanding the behavior of the expression.