Problem 56 - Algebra 2

Question

REVIEW \(\frac{x+2}{x+3}+\frac{4}{x^{2}+x-6}=\) $$ \begin{array}{ll}{\mathbf{F} \frac{-3 x-9}{x^{2}+x-6}} & {\mathbf{H} \frac{x^{2}}{x^{2}+x-6}} \\ {\mathbf{G} \frac{x^{2}-3 x-24}{x^{2}+x-6}} & {\mathbf{J} \frac{x^{2}+x-1}{x^{2}+x-6}}\end{array} $$

Step-by-Step Solution

Answer

H: \(\frac{x^2}{x^2+x-6}\).

Step 1: Factor the Denominator

Begin by factoring the quadratic expression in the second fraction's denominator: \(x^2 + x - 6\). To factor, find two numbers that multiply to \(-6\) and add to \(1\). These numbers are \(3\) and \(-2\). Thus, \(x^2 + x - 6 = (x + 3)(x - 2)\).

Step 2: Rewrite the Expression

The given expression becomes: \(\frac{x+2}{x+3} + \frac{4}{(x+3)(x-2)}\). Notice that the first fraction's denominator is a factor of the second fraction's denominator.

Step 3: Find a Common Denominator

The common denominator of both fractions is \((x+3)(x-2)\). Adjust the first fraction to this common denominator: \(\frac{x+2}{x+3} = \frac{(x+2)(x-2)}{(x+3)(x-2)}\).

Step 4: Expand the Numerator of the First Fraction

Expand \((x+2)(x-2)\) using the distributive property: \((x+2)(x-2) = x^2 - 2x + 2x - 4 = x^2 - 4\). So the first fraction becomes \(\frac{x^2 - 4}{(x+3)(x-2)}\).

Step 5: Add the Fractions

Add the two fractions: \(\frac{x^2 - 4}{(x+3)(x-2)} + \frac{4}{(x+3)(x-2)} = \frac{x^2 - 4 + 4}{(x+3)(x-2)}\).

Step 6: Simplify the Numerator

Combine the terms in the numerator: \(x^2 - 4 + 4 = x^2\). Thus, the expression simplifies to \(\frac{x^2}{(x+3)(x-2)}\).

Step 7: Compare with Answer Choices

Looking at the answer choices, we see that this matches option \(\mathbf{H}\), which is \(\frac{x^2}{x^2+x-6}\).

Key Concepts

Common Denominator

When working with fractions, finding a common denominator is crucial for operations like addition or subtraction. This means determining a common base for all fractions involved, allowing you to combine them easily.

Let's consider the problem:

• We have two fractions: \( \frac{x+2}{x+3} \) and \( \frac{4}{x^2+x-6} \).
• The second denominator was factored as \((x+3)(x-2)\). This helps us identify \((x+3)(x-2)\) as the common denominator.
• The first fraction's denominator \( (x+3) \) is already part of this common denominator, simplifying our task.
• To achieve a common base, we adjust the first fraction like so: \( \frac{(x+2)(x-2)}{(x+3)(x-2)} \).

This approach makes manipulating the fractions straightforward. Practice identifying and using common denominators to streamline your calculations with rational expressions.

Rational Expressions

Rational expressions are similar to fractions but involve polynomials in their numerators and denominators. Just as with simple fractions, operations on rational expressions require careful handling of polynomials.

Consider these key points:

• A rational expression is written as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
• When adding or subtracting, ensure a common denominator is found to facilitate combination.
• Be sure to factor any polynomial expressions to simplify the process, as seen in our exercise where \(x^2 + x - 6\) was factored to \((x+3)(x-2)\).
• Once a common denominator is achieved, operations can then be performed on the numerators.

Handling rational expressions efficiently involves good factoring skills and recognizing common patterns.

Numerator Expansion

When dealing with rational expressions, expanding numerators is often necessary to perform operations like addition. This involves using algebraic techniques to simplify or manipulate the expressions involved.

Looking at our example:

• We began with \(\frac{(x+2)(x-2)}{(x+3)(x-2)}\): To simplify, expand the numerator.
• Apply the distributive property:
  • \((x+2)(x-2) = x^2 - 2x + 2x - 4 = x^2 - 4\)
• The expression becomes \(\frac{x^2 - 4}{(x+3)(x-2)}\).
• With expanded numerators, similar terms can be combined, as evident when adding fractions: \(x^2 - 4 + 4 = x^2\).

Expanding numerators is a vital step, allowing fractions to be simplified or added seamlessly. By mastering this, complex rational expressions become more manageable.