Problem 40 - College Algebra

Question

Add or subtract as indicated. $$ \frac{x^{2}-4 x}{x^{2}-x-6}-\frac{x-6}{x^{2}-x-6} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{x^{2}-5x + 6}{x^{2}-x-6}\).

Step 1: Identify the Same Denominator

Looking at these two fractions, we can observe that they both have the same denominator which is \(x^{2}-x-6\). It's important in the subtraction of fractions.

Step 2: Combine the Numerators

Since the denominators of both fractions are the same, we can simply subtract the numerators: \( (x^{2}-4x) - (x-6) \). The subtraction in the numerator should be treated as adding the opposite. In other words, we rewrite as \(x^{2}-4x - x + 6\).

Step 3: Simplify the Numerator

Now, by combining like terms in the numerator, we simplify the fraction. That would look like this \(x^{2}-5x + 6 \) over the original denominator \(x^{2}-x-6 \). Now, we have the simplified fraction: \(\frac{x^{2}-5x + 6}{x^{2}-x-6}\).

Step 4: Check for further simplification

We check whether the numerator and denominator can be factored and if they can cancel out any common factors. The given fraction is already in its simplest form.

Key Concepts

Subtracting Rational Expressions

Subtracting rational expressions is similar to subtracting regular fractions. The key to doing this efficiently is ensuring that both expressions involved share the same denominator. Rational expressions are just fractions where the numerator and/or the denominator are polynomials. To subtract these expressions:

• Ensure they have a common denominator.
• Subtract the numerators while keeping the denominator the same.
• Simplify the resulting expression if possible.

Just like with regular numbers, the denominator dictates how the numerators interact. If the denominators differ, you must first find a common one, much like finding a common denominator with simple fractions.

Like Denominators

When working with rational expressions, having like denominators makes subtraction much easier. Think of the denominator like a foundation; when each expression has the same foundation, you don't need to make any adjustments to the expressions before combining or subtracting them. In the context of our problem:

• Both expressions have the denominator \(x^{2}-x-6\).
• Since they are the same, we can directly subtract the numerators.

Once the subtraction is complete, the denominator remains unchanged, maintaining the consistency of your rational expression. This simplifies the subtraction process and allows for straightforward manipulation of the numerators.

Simplifying Expressions

Simplifying expressions is a crucial part of managing and maintaining mathematical problems more effectively. Once you have subtracted the numerators of rational expressions with like denominators, simplifying can help reveal a more reduced form, eye-friendly for further operations if needed. To simplify:

• Combine like terms in the numerator, as seen in our example: \(x^{2} - 5x + 6\).
• Check whether the resulting expression can be factored.
• If applicable, reduce by canceling any common factors between the numerator and the denominator.

Simplifying not only makes the expression cleaner but can help in understanding relationships between mathematical components and structures more intuitively.

Factoring Polynomials

Factoring polynomials is the process of breaking down a complex polynomial into a product of simpler polynomials. This process is essential for simplifying expressions further and sometimes necessary in reducing rational expressions.In our example, after obtaining the numerator \(x^{2} - 5x + 6\), you would:

• Attempt to factor it into simpler polynomials. However, if no common factor exists between the numerator and denominator, as is the case here, further simplification doesn't apply.

Factoring comes into play primarily when you are looking at the possibility of canceling out terms in the numerator and denominator. Hence, a deep understanding of factoring can make managing complex expressions more approachable and easier to grasp, leaving you with a simplified form that's often easier to interpret or solve in further steps.Always remember, effective use of factoring requires practice and a thorough check of both numerator and denominator possibilities for simplification.