Problem 45 - Algebra 2

Question

What are the restrictions on \(x\) when \(\frac{x^{2}-x-2}{x^{2}-9}\) is divided by \(\frac{x-8}{x^{2}+10 x+25} ?\) $$ \begin{array}{ll}{F . x \neq-3 \text { or }-5} & {\text { G. } x \neq 3,-3, \text { or }-5} \\ {\text { H. } x \neq 3,-3,-5, \text { or } 8} & {\text { J. } x \neq 2,9,8, \text { or }-25}\end{array} $$

Step-by-Step Solution

Answer

H. \(x \neq 3,-3,-5\)

Step 1: Finding roots of the main fraction

Firstly, solve the quadratic equation in the denominator of the main fraction, \(x^{2}-9=0\). This quadratic equation can be factored into \((x-3)(x+3)=0\). By the zero product property, if a product of factors is zero, then at least one of the factors must be zero, leading to the solutions \(x=-3\) and \(x=3\). These are the roots of the quadratic equation, and the values that return the denominator of the main fraction to zero.

Step 2: Finding roots of the divider fraction

Secondly, let's solve the quadratic equation in the denominator of the divider fraction, \(x^{2}+10x+25=0\). Factoring the quadratic gives \((x+5)^2 = 0\). Then, using the zero product property again, the root that makes the denominator of the divider fraction equal to zero is \(x=-5\).

Step 3: Determine the restrictions on 'x'

The root of the linear function in the numerator of the divisor must be found, \(x-8=0\). That results in \(x=8\). But notice that \(x=8\) doesn’t make the divisor zero, therefore, it’s not a restriction. To conclude, \(x\) cannot equal to the roots found earlier, namely \(x=-3, x=3, x=-5\). So, in this case, the restrictions on \(x\) are \(x \neq 3, -3, -5\).

Key Concepts

Understanding Restrictions in Rational Expressions

When dealing with rational expressions, restrictions are crucial to ensure the expression remains valid. The restrictions arise because the value of the denominator cannot be zero. This is fundamental in mathematics because division by zero is undefined. To find restrictions:

• Identify the denominators in the rational expression.
• Solve each denominator equation such that it equals zero to find potential values of \( x \) that would make the denominator zero.
• Exclude these values from the domain of the expression, as they create invalid scenarios.

These exclusions are the restrictions on \( x \). Recognizing these ensures the rational expression is properly defined.

Quadratic Equations in Rational Expressions

Quadratic equations often appear in the denominators of rational expressions, influencing the restrictions for these expressions. A quadratic equation generally takes the form \( ax^2 + bx + c = 0 \). Solving this equation involves finding the roots or the values of \( x \) that satisfy it.Quadratic equations may be solved using:

• Factoring, when the equation is easily breakable into products, like \((x-3)(x+3)=0\).
• Quadratic formula, especially useful when factoring is complex or impractical: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• Completing the square, which reformulates the equation into a perfect square trinomial.

The roots found indicate potential denominator zeros, hence leading to the expressions' restrictions.

Importance of the Denominator in Rational Expressions

In rational expressions, the denominator plays a vital role in establishing the validity of the expression. As a fundamental rule, any value of \( x \) that results in a zero denominator must be restricted from the set of possible solutions.The denominator impacts rational expressions by:

• Determining the domain of the expression.
• Influencing any simplification of the expression, for instance, canceling common factors between the numerator and denominator.
• Affecting the behavior and continuity of the function represented by the expression.

Understanding the denominator's role helps in managing complications that arise from division by zero, maintaining mathematical integrity in calculations.