Problem 5 - College Algebra

Question

Find all numbers that must be excluded from the domain of each rational expression. $$ \frac{x-1}{x^{2}+11 x+10} $$

Step-by-Step Solution

Answer

The numbers that ought to be excluded from the domain of the rational expression are \(x = -1\) and \(x = -10\).

Step 1: Recognize the Rational Expression and Its Domain

First, recognize that the expression is a ratio of two polynomials, which makes it a rational expression. The domain of a rational expression is all the real numbers except those which make the denominator equal to zero.

Step 2: Set the Denominator Equal to Zero

To locate these values, the denominator \(x^2 + 11x + 10\) is set equal to zero, yielding an equation to solve: \(x^2 + 11x + 10 = 0\).

Step 3: Solve for x

The quadratic can be factored to \((x + 1)(x + 10) = 0\). Setting eachfactor equal to zero gives the roots \(x = -1\) and \(x = -10\). These are the values to be excluded from the domain.

Key Concepts

Understanding Rational Expressions

Rational expressions are a central part of algebra that you will encounter often. As the name suggests, a rational expression is a fraction where both the numerator and the denominator are polynomials. For example, in the expression \( \frac{x-1}{x^2 + 11x + 10} \), \( x-1 \) is the numerator, and \( x^2 + 11x + 10 \) is the denominator. Understanding rational expressions involves recognizing their structure and knowing how to manipulate them effectively.

It is crucial to remember that the denominator of a rational expression cannot be zero. If the denominator equals zero, the expression becomes undefined, which is why finding the domain of a rational expression implies finding all values of the variable that make the denominator non-zero.

Role of Polynomials in Rational Expressions

Polynomials are multi-term expressions which involve terms in the form of \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\), where each \(a\) represents coefficients, and \(x\) is a variable. The denominator of a rational expression is often a polynomial, making it a key component in determining the expression's domain. In our example, the polynomial \(x^2 + 11x + 10\) is the denominator.

A significant aspect of working with polynomials in rational expressions is recognizing the situations where they might equal zero, as these values are excluded from the domain. Factoring polynomials is a common way to simplify these expressions and find crucial information, like the roots.

Finding Roots of Equations

When finding the domain of rational expressions, a critical step is solving equations that set the denominator equal to zero. These solutions give us the values to be excluded from the domain. To find these roots, you can often apply methods from algebra, such as factoring, graphing, or using the quadratic formula.

For instance, setting \(x^2 + 11x + 10 = 0\) helps in finding the domain exclusions of \( \frac{x-1}{x^2 + 11x + 10} \). By factoring, you can simplify the equation into linear terms to easily find its roots. For this example, the polynomial factors to \((x+1)(x+10)=0\), thus the roots are \(x = -1\) and \(x = -10\). These roots must not be included in the domain of the rational expression.

Factoring Quadratics

Factoring is an essential skill in algebra that simplifies expressions and solves equations. Specifically, factoring quadratics—polynomials of the form \(ax^2 + bx + c\)—is invaluable when working with equations like \(x^2 + 11x + 10 = 0\). To factor a quadratic, you seek two numbers that multiply to give \(ac\) (the product of the coefficient of \(x^2\) and the constant term \(c\)) and add to give \(b\) (the coefficient of \(x\)).

In our case, we find that 1 and 10 are those numbers because \(1 \times 10 = 10\) and \(1 + 10 = 11\). This means the quadratic can be expressed as \((x+1)(x+10)=0\). Factoring turns the quadratic equation into a simpler form that makes it easier to solve for zero and helps in identifying the excluded values from a rational expression's domain.