Problem 12 - Algebra 2

Question

Simplify each expression. $$ \frac{x}{x+1}+\frac{3}{x^{2}-4 x-5} $$

Step-by-Step Solution

Answer

Simplified expression: \( \frac{x^2 - 5x + 3}{(x + 1)(x - 5)} \).

Step 1: Factor the Denominator

We need to simplify the expression by combining the fractions. Begin by factoring the second denominator, which is a quadratic expression. \[ x^2 - 4x - 5 = (x - 5)(x + 1) \]The factors of -5 that sum to -4 are -5 and 1.

Step 2: Find a Common Denominator

The common denominator for the fractions \( \frac{x}{x+1} \) and \( \frac{3}{(x-5)(x+1)} \) is \((x + 1)(x - 5)\).

Step 3: Rewrite Each Fraction

Rewrite each fraction with the common denominator. For \( \frac{x}{x+1} \), multiply the numerator and the denominator by \( (x - 5) \):\[ \frac{x(x - 5)}{(x + 1)(x - 5)} \] For \( \frac{3}{(x-5)(x+1)} \) it remains the same:\[ \frac{3}{(x-5)(x+1)} \]

Step 4: Combine the Fractions

Add the fractions now that they have the common denominator:\[ \frac{x(x - 5) + 3}{(x + 1)(x - 5)} \]

Step 5: Simplify the Numerator

Simplify the numerator by distributing and combining like terms:\[ x(x-5) = x^2 - 5x \] Add the 3:\[ x^2 - 5x + 3 \]

Step 6: Final Simplified Expression

The final simplified expression is:\[ \frac{x^2 - 5x + 3}{(x + 1)(x - 5)} \]

Key Concepts

Factoring Quadratics

When dealing with rational expressions, factoring quadratics is a crucial step. Quadratic expressions, such as \( x^2 - 4x - 5 \), often appear in the denominator of a fraction. To factor them, you are looking for two numbers that multiply to give the constant term, in this case -5, and add to give the linear coefficient, -4.
For the expression \( x^2 - 4x - 5 \), these numbers are -5 and 1. This allows us to express the quadratic as \((x - 5)(x + 1)\). This factored form is essential because it helps in finding a common denominator and simplifying the expression.

• Check the factors of the constant term.
• Ensure their sum equals the linear coefficient.
• Rewrite the quadratic as the product of two binomials.

Common Denominator

Finding a common denominator is key to combining fractions with different denominators. It allows us to rewrite the fractions so that they have the same denominator, making addition or subtraction possible.
For example, with the fractions \( \frac{x}{x+1} \) and \( \frac{3}{(x-5)(x+1)} \), the denominators are \( x+1 \) and \((x - 5)(x + 1)\). The simplest way to ensure both fractions have the same denominator is to find the least common multiple of both:

• Identify each distinct factor from all denominators.
• Multiply these factors together to form the common denominator.
• In this case, the common denominator is \((x+1)(x-5)\).

By using this common denominator, the fractions can be rewritten and combined.

Simplifying Expressions

Simplifying expressions involves manipulating them into their simplest or most reduced form. After setting a common denominator for the fractional expressions, simplifying the expression means pruning down the numerator and/or the denominator to its simplest encompassing terms.
Here's how it's done:

• Once the fractions have been combined, look at the new numerator. In this case, it becomes \( x(x - 5) + 3 \).
• Distribute any factored terms across sums and/or products. Here, distribute the \( x \) across \( (x - 5) \) to get \( x^2 - 5x \).
• Combine like terms if necessary. This results in \( x^2 - 5x + 3 \).

The goal is to end up with a fraction where both numerator and denominator are expressed in the simplest form possible.

Combining Fractions

Combining fractions is the process of taking two or more fractions with a common denominator and combining their numerators. This is essential for any algebraic manipulation involving fractions because it allows you to condense expressions into a more manageable form.
With the fractions \( \frac{x}{x+1} \) and \( \frac{3}{(x-5)(x+1)} \) with a common denominator \((x + 1)(x - 5)\), the addition becomes:

• Rewriting \( \frac{x}{x+1} \) as \( \frac{x(x-5)}{(x+1)(x-5)} \) establishes the common denominator.
• Add the numerators: \( x(x-5) + 3 \).
• The resulting expression is \( \frac{x^2 - 5x + 3}{(x + 1)(x - 5)} \).

Remember to always check if the expression can be simplified further after combination.