Problem 70 - Algebra A Combined Function

Question

Write two rational expressions with the same denominator whose difference is \(\frac{x-7}{x^{2}+1}\).

Step-by-Step Solution

Answer

\( \frac{x}{x^2+1} \) and \( \frac{7}{x^2+1} \) are the expressions.

Step 1: Understand the Problem

We need to find two rational expressions that have the same denominator. Their difference should equal \( \frac{x-7}{x^2+1} \). The expressions will look like \( \frac{A}{x^2+1} \) and \( \frac{B}{x^2+1} \).

Step 2: Set Up the Equation

Based on the requirement, we need \( \frac{A}{x^2+1} - \frac{B}{x^2+1} = \frac{x-7}{x^2+1} \). This equation simplifies to \( \frac{A - B}{x^2+1} = \frac{x-7}{x^2+1} \).

Step 3: Equate Numerators

Since the denominators are equal, we equate the numerators: \[ A - B = x - 7 \]Our task is to find any two numbers \( A \) and \( B \) such that their difference equals \( x-7 \).

Step 4: Choose Possible Numerators

Let's choose \( A = x \) and \( B = 7 \). Substituting these values into our equation \( A - B = x - 7 \) becomes \( x - 7 = x - 7 \), which holds true.

Step 5: Write the Rational Expressions

We now write the rational expressions as:\[ \frac{x}{x^2+1} \] and \[ \frac{7}{x^2+1} \] These expressions satisfy the given condition.

Key Concepts

Common Denominator

In the world of rational expressions, a common denominator is key to many operations, particularly when adding or subtracting fractions. A denominator is the part of the fraction that resides beneath the fraction bar. For rational expressions, it typically contains variables and/or constants. Finding a common denominator means that the expressions you wish to operate on share the same bottom part - making it much easier to manipulate them.

Here’s why it is necessary:

• Allows for Simplification: By having the same denominator, you can easily combine or manipulate the fractions.
• Facilitates Comparison: A common denominator makes it possible to compare fractions directly.
• Makes the Process Streamlined: It simplifies complex calculations when moving terms around or solving equations.

In our case, both rational expressions must have the denominator \(x^{2}+1\) to work seamlessly in the subtraction process we’re looking at. By maintaining this shared denominator, you focus solely on the numerators in subsequent steps.

Subtracting Rational Expressions

When subtracting rational expressions, identifying a common denominator is your first goal. Once achieved, the process mirrors subtraction in basic arithmetic but applied to more complicated symbolic fractions.

In simpler terms, here's the process:

• Ensure both expressions have the same denominator. This step was achieved previously with \(x^{2}+1\).
• Focus purely on the numerators. Since the denominators are identical, you only need to subtract the numerators.
• Form the new expression by combining the numerators under the common denominator.

Remember, you subtract rational expressions by treating them as though they are numbers; only when denominators are unified do you proceed to handle the numerators. In our exercise, this allows the subtraction: \( \frac{A}{x^2+1} - \frac{B}{x^2+1} = \frac{A - B}{x^2+1} \). This direct approach while intuitive is what typically confuses students on the topic; hence why building a strong conceptual understanding is vital.

Equating Numerators

Equating numerators is a direct consequence of having a common denominator when subtracting rational expressions. It simplifies the task by reducing it to a simpler arithmetic problem involving only the numerators.

Here's how it's done:

• Start by setting the numerators of the equivalent expressions equal to each other. This removes the denominator concern entirely since they match.
• Solve the equation formed by the numerators alone. In our case, this means solving \(A - B = x - 7\).
• Select appropriate values that satisfy this equation. Creativity in choice can reveal multiple solutions fitting the same criteria.

For this problem, choosing \(A = x\) and \(B = 7\) satisfies \(A - B = x - 7\) perfectly. Ultimately, equating numerators aids in managing complex expressions by breaking them down into relatable sections, thus enhancing understanding and simplification.