Problem 46 - Algebra 2

Question

Let \(f(x), g(x), h(x),\) and \(k(x)\) be rational expressions. Explain how to find all the restrictions for \(\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}\)

Step-by-Step Solution

Answer

The restrictions on \(x\) in the expression \(\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}\) come from the conditions \(g(x) \neq 0\) and \(h(x) \neq 0\). Solve these equations to get all the restrictions on \(x\).

Step 1: Write the Division as Multiplication

First, replace the division of two fractions with the multiplication by inverting the second fraction. Write \(\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}\) as \(\frac{f(x)}{g(x)} \times \frac{k(x)}{h(x)}\).

Step 2: Find the Restrictions of the Rational Expressions

Find the restrictions of the numerator and denominator of our new rational expression, which comes from the conditions \(g(x) \neq 0\) and \(h(x) \neq 0\). Solve these equations to find the values of \(x\) which would make these equal to zero. These values are the restrictions on \(x\) in our expression.

Step 3: Combine the Restrictions

Combine all the restrictions obtained in step 2. These are the restrictions on \(x\) for the original expression \(\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}\). If any restrictions repeat, just count them once.

Key Concepts

Division of Rational Expressions

Division of rational expressions can seem complicated at first, but it becomes easier once you realize it's closely related to what we do with numbers. When dividing two fractions, you change the division problem into a multiplication problem by flipping the second fraction. The same happens with rational expressions. For example, when you have the division \(\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}\), you can rewrite it as a multiplication: \(\frac{f(x)}{g(x)} \times \frac{k(x)}{h(x)}\). This step is crucial and simplifies the process, making it more straightforward to solve or simplify the expression.

Restricting Values

Whenever you are working with rational expressions, you must identify values of the variable that make the expression undefined. These are known as restricting values. The expression is undefined when the denominator of any fraction equals zero. In the division problem, \(\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}\), you switch to multiplication as \(\frac{f(x)}{g(x)} \times \frac{k(x)}{h(x)}\). To find the restrictions, check the original denominators, \(g(x)\) and \(h(x)\).

• Set each denominator equal to zero.
• Solve the equation for \(x\) to find the restricting values.

These values of \(x\) are the restrictions because if \(x\) takes any of these values, the denominator becomes zero, making the expression undefined. Remembering this step ensures accuracy in solving rational expressions.

Multiplication of Rational Expressions

Once you've converted division into multiplication, handling rational expressions becomes more manageable. Consider the expression \(\frac{f(x)}{g(x)} \times \frac{k(x)}{h(x)}\). When multiplying, keep these key points in mind:

• Multiply the numerators together.
• Multiply the denominators together.

This follows the same procedures as multiplying simple fractions. After multiplying, simplify the expression by canceling any common factors between the numerator and the denominator if possible. Simplification makes the expression neater and often reveals restrictions or further insights into the values that \(x\) can take. Always keep the restrictions found in mind, since they influence the validity of your simplified expression.