Problem 21 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{2}{r^{2}-3 r-10}+\frac{r}{r^{2}-3 r-10}\)

Step-by-Step Solution

Answer

\( \frac{2 + r}{(r - 5)(r + 2)} \) is the simplified expression.

Step 1: Identify the Common Denominator

Both fractions have the same denominator, which is \(r^2 - 3r - 10\). Therefore, they can be combined directly since they already share the common denominator.

Step 2: Combine the Fractions

Since the denominators are the same, add the numerators of the fractions directly:\[\frac{2}{r^2 - 3r - 10} + \frac{r}{r^2 - 3r - 10} = \frac{2 + r}{r^2 - 3r - 10}\]

Step 3: Simplify the Numerator

The numerator \(2 + r\) is already in its simplest form, so this fraction is already simplified as much as possible given the provided terms.

Step 4: Factor the Denominator

To ensure that the expression is fully simplified, determine if the denominator can be factored:The expression \( r^2 - 3r - 10 \) factors into \((r - 5)(r + 2)\). Thus the expression becomes:\[\frac{2 + r}{(r - 5)(r + 2)}\]

Step 5: Check for Further Simplification

Examine if the numerator \(2 + r\) can cancel with any factor in the denominator. Since \(2 + r\) does not match any factor, the fraction is in its simplest form.

Key Concepts

Adding Fractions

Adding fractions might seem challenging at first, but it can become straightforward once you understand the main rule: fractions can only be added when they have the same denominator. In algebra, adding fractions works the same way as with numbers. The key is to focus on the denominator, which is the part of the fraction below the fraction bar. Let's explore how to do this:

To add

• First, ensure that all fractions involved have the same denominator. This is crucial because fractions with different denominators cannot be directly added – their values represent different "sizes," just like different units in a measurement.
• Once you confirm they have a common denominator, keep this denominator the same and simply add the numerators, which are the top parts of the fractions.
• Write the sum of the numerators over the common denominator. This gives you a new fraction that combines the values of the original fractions:\(rac{a}{d} + rac{b}{d} = \frac{a+b}{d}\)

This step is straightforward when the fractions already have a common denominator, as seen in our example \(\frac{2}{r^2 - 3r - 10} + \frac{r}{r^2 - 3r - 10}\). Since both fractions share the same denominator, you can directly add the numerators, making the process both easy and efficient.

Common Denominator

To successfully add fractions, you must determine the common denominator. This denominator is essentially a shared base that allows fractions to be added directly. But what if your fractions have different denominators? Here's how to get them to share one:

• Find the least common multiple (LCM) of the denominators. The LCM is the smallest number that can be divided by each of the denominators without a remainder.
• Once you have the LCM, adjust each fraction so that its denominator matches this common denominator. This is done by multiplying both the top (numerator) and bottom (denominator) by the same number to adjust without changing the fraction's value.
• With the denominators aligned, proceed to add the fractions using the method already described.

In the exercise, we are fortunate because the two fractions already share a denominator: \(r^2 - 3r - 10\). This pre-existing common denominator allows for immediate addition without any extra adjustments, making the process quick and eliminating any need to calculate an LCM.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making them as concise as possible. Once fractions are added, it's often necessary to "tidy up" the expression. Here's the approach:

• Simplify the Numerator: Combine like terms and arrange the terms in a standard order, usually descending by degree. In our example, this process involves simply recognizing that the numerator \(2 + r\) is already as simple as it can be.
• Factor the Denominator: Analyze the expression to see if it can be rewritten as a product of simpler expressions. In our exercise, the quadratic \(r^2 - 3r - 10\) factors neatly into \((r - 5)(r + 2)\). Factoring reveals the structure of the expression and can reveal further opportunities for simplification.
• Cancel Common Factors: Look for terms that appear in both the numerator and the denominator. If any terms are identical, they can "cancel out" each other, further simplifying the fraction. In this case, the numerator and denominator do not share factors, indicating that the fraction is fully simplified.

Simplifying expressions is a powerful tool in algebra that not only aids in solving problems but also helps in extending understanding of algebraic structures and relationships.