Problem 58 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{4}{x^{2}+3 x+2}+\frac{9}{x^{2}+6 x+8} $$

Step-by-Step Solution

Answer

Question: Perform the indicated operation and simplify: $$\frac{4}{x^2 + 3x + 2}+\frac{9}{x^2 + 6x + 8}$$ Answer: $$\frac{13x + 25}{(x+1)(x+2)(x+4)}$$

Step 1: Factor the denominators

First, we need to factor the two quadratic expressions in the denominators: $$ x^2 + 3x + 2 = (x+1)(x+2) $$ $$ x^2 + 6x + 8 = (x+2)(x+4) $$

Step 2: Find the LCM of denominators

Now, we will find the LCM of the two factored denominators: LCM \(= (x+1)(x+2)(x+4)\)

Step 3: Rewrite the fractions with the LCM as denominator

Next, we rewrite both fractions with the LCM as the common denominator: $$ \frac{4}{(x+1)(x+2)}+\frac{9}{(x+2)(x+4)} \Rightarrow \frac{4(x+4)+9(x+1)}{(x+1)(x+2)(x+4)} $$

Step 4: Simplify the numerator and combine the fractions

Now, we will simplify the numerator of the combined fraction and then combine it: $$ \frac{4(x+4)+9(x+1)}{(x+1)(x+2)(x+4)} = \frac{4x + 16 + 9x + 9}{(x+1)(x+2)(x+4)} $$ $$ = \frac{13x + 25}{(x+1)(x+2)(x+4)} $$

Step 5: Write the final answer

The result after performing the indicated operation is: $$ \frac{13x + 25}{(x+1)(x+2)(x+4)} $$

Key Concepts

Factoring Quadratics

Factoring quadratics is a vital step in simplifying algebraic expressions, particularly when dealing with algebraic fractions. It involves breaking down a quadratic expression, such as \(x^2 + 3x + 2\), into a product of linear factors. This makes further operations more straightforward.

To factor a quadratic expression, identify two numbers that multiply to the constant term (last number of the quadratic) and add up to the linear coefficient (the middle term). For example:

• In \(x^2 + 3x + 2\), we need two numbers that multiply to 2 and add to 3 — these numbers are 1 and 2.
• This results in the factored form \((x + 1)(x + 2)\).

Factoring allows us to see the roots or zeroes of the quadratics, which are \(x = -1\) and \(x = -2\) in this example. Recognizing these roots is essential when solving equations or simplifying expressions with common factors.

Least Common Multiple

Finding the least common multiple (LCM) is a key step when working with algebraic fractions. It is used to create a common denominator, which is crucial for adding, subtracting, or comparing fractions. The LCM of two or more expressions is the smallest expression that is a multiple of each component.

To find the LCM of the denominators \((x+1)(x+2)\) and \((x+2)(x+4)\), initially factor each component as much as possible. Then, take each unique factor that appears in any of the expressions, raised to the highest power that appears in any:

• From \((x+1)(x+2)\) and \((x+2)(x+4)\), extract the factors \((x+1)\), \((x+2)\), and \((x+4)\).
• This gives us the LCM: \((x+1)(x+2)(x+4)\).

With the LCM, we can rewrite algebraic fractions so they share this common denominator, making addition and subtraction possible. It simplifies the processing of fractions, helping us focus on the numerators.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form while retaining equivalence. When simplifying algebraic fractions, especially after finding a common denominator, we aim to combine like terms and reduce ease of interpretation.

Consider the expression \( \frac{4(x+4) + 9(x+1)}{(x+1)(x+2)(x+4)} \). We simplify the numerator by distributing and combining like terms:

• Distribute: \(4(x+4) = 4x + 16\) and \(9(x+1) = 9x + 9\).
• Combine: \(4x + 9x = 13x\) and \(16 + 9 = 25\).

As a result, the simplified expression is \( \frac{13x + 25}{(x+1)(x+2)(x+4)} \). Emphasizing steps like these facilitates understanding of how terms relate and interact, ultimately conveying how the original, more complex expressions can be represented more succinctly.