Problem 56 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{r}{r^{2}+5 r+6}-\frac{2}{r^{2}+3 r+2} $$

Step-by-Step Solution

Answer

The simplified difference is \(\frac{r - 3}{(r + 1)(r + 3)}\).

Step 1: Factor the Denominators

First, factor the denominators of the two fractions. The denominator of the first fraction is \( r^2 + 5r + 6 \), which factors into \((r + 2)(r + 3)\). The denominator of the second fraction is \( r^2 + 3r + 2 \), which factors into \((r + 1)(r + 2)\).

Step 2: Determine the Least Common Denominator (LCD)

To subtract the fractions, we need to combine them over a common denominator. The least common denominator of \((r + 2)(r + 3)\) and \((r + 1)(r + 2)\) is \((r + 1)(r + 2)(r + 3)\).

Step 3: Express Each Fraction with the LCD

Re-write each fraction so they both have the LCD as the denominator. For the first fraction, multiply the numerator and denominator by \(r + 1\), yielding \(\frac{r(r + 1)}{(r + 1)(r + 2)(r + 3)}\). For the second fraction, multiply the numerator and the denominator by \(r + 3\), resulting in \(\frac{2(r + 3)}{(r + 1)(r + 2)(r + 3)}\).

Step 4: Subtract the Fractions

Now subtract the fractions using their common denominator: \(\frac{r(r + 1) - 2(r + 3)}{(r + 1)(r + 2)(r + 3)}\).

Step 5: Simplify the Numerator

Expand and simplify the numerator: \(r(r + 1) - 2(r + 3) = r^2 + r - 2r - 6 = r^2 - r - 6\).

Step 6: Present the Simplified Result

The simplified expression for the entire operation is \(\frac{r^2 - r - 6}{(r + 1)(r + 2)(r + 3)}\). Now check if the numerator can be factored. It factors into \((r - 3)(r + 2)\).

Step 7: Further Simplify by Canceling Common Factors

Notice that \((r + 2)\) appears in both the numerator and denominator, so they cancel out. The final simplified expression is \(\frac{r - 3}{(r + 1)(r + 3)}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that allows us to simplify expressions and solve equations. It involves writing a polynomial as a product of its factors, which are simpler polynomials. In this exercise, we factored the quadratic polynomials in the denominators of each fraction. Let's explore how to do this effectively.

Polynomials like the denominators of our fractions, \(r^2 + 5r + 6\) and \(r^2 + 3r + 2\), are expressions that can often be factored by looking for two numbers that multiply to the constant term and add to the coefficient of the linear term. For instance:

• For \(r^2 + 5r + 6\), we look for two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3, so we factor it into \((r + 2)(r + 3)\).
• For \(r^2 + 3r + 2\), we need two numbers that multiply to 2 and add to 3. These numbers are 1 and 2, allowing us to factor it as \((r + 1)(r + 2)\).

Factoring a polynomial allows us to see the zeros of the function represented by the polynomial. In this context, it helps us find a common ground—or, literally, a common denominator—when working with fractions.

Least Common Denominator

Understanding and finding the least common denominator (LCD) is a vital step when adding or subtracting fractions. The LCD is the smallest expression that all denominators can divide without a remainder, much like the least common multiple for numbers.

In our example, we discovered the LCD by analyzing the factored forms of the denominators. We note common terms and those not shared to construct the lowest shared structure:

• The first denominator \((r + 2)(r + 3)\).
• The second denominator \((r + 1)(r + 2)\).

Notice that \((r + 2)\) appears in both, \(r + 1\) from the second, and \(r + 3\) from the first are distinct factors. Thus, the LCD becomes \((r + 1)(r + 2)(r + 3)\). Having a common denominator allows the fractions to be combined thus facilitating subtraction, addition, or comparison of the fractions involved. Always identify and use the least common denominator to maintain the integrity and simplicity of your resulting expression.

Simplifying Expressions

Simplifying expressions is about making them as compact and manageable as possible. After rewriting fractions with the least common denominator, the next step involves simplifying both numerator and denominator.

Once we have our common denominator, let's focus on simplifying the combined numerator:

• Expand and simplify: \(r(r + 1) - 2(r + 3)\) becomes \(r^2 + r - 2r - 6 = r^2 - r - 6\).
• Factor the resulting expression if possible. Here, \(r^2 - r - 6\) is factored as \((r - 3)(r + 2)\).

Next, observe any common factors in the numerator and the common denominator—which here include \((r + 2)\)—and cancel them out.

The simplified fraction becomes \(\frac{r - 3}{(r + 1)(r + 3)}\). By canceling common terms properly, the expression simplifies, which is easier to interpret or further compute. Simplification also aids in identifying potential equalities or inequalities in mathematical contexts, ensuring clarity and precision in calculations.