Problem 62 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{2 k}{k^{2}+4 k+3}+\frac{3 k}{k^{2}+5 k+6}$$

Step-by-Step Solution

Answer

The sum is \(\frac{5k^2 + 7k}{(k + 1)(k + 2)(k + 3)}\).

Step 1: Factor the Denominators

First, we need to factor each of the quadratic expressions in the denominators. For the first denominator, \(k^2 + 4k + 3\), we factor it into \((k + 1)(k + 3)\). For the second denominator, \(k^2 + 5k + 6\), we factor it into \((k + 2)(k + 3)\).

Step 2: Find a Common Denominator

The common denominator is the least common multiple of \((k + 1)(k + 3)\) and \((k + 2)(k + 3)\), which is \((k + 1)(k + 2)(k + 3)\).

Step 3: Rewrite Each Fraction with the Common Denominator

To adjust the first fraction, \(\frac{2k}{(k + 1)(k + 3)}\), we multiply the numerator and denominator by \(k + 2\), so it becomes \(\frac{2k(k + 2)}{(k + 1)(k + 2)(k + 3)}\). For the second fraction, \(\frac{3k}{(k + 2)(k + 3)}\), we multiply by \(k + 1\), resulting in \(\frac{3k(k + 1)}{(k + 1)(k + 2)(k + 3)}\).

Step 4: Combine the Fractions

Now that both fractions have the same denominator, combine them: \(\frac{2k(k + 2) + 3k(k + 1)}{(k + 1)(k + 2)(k + 3)}\).

Step 5: Simplify the Numerator

Expand \(2k(k + 2)\) to get \(2k^2 + 4k\) and \(3k(k + 1)\) to get \(3k^2 + 3k\). Add these to get \(2k^2 + 4k + 3k^2 + 3k = 5k^2 + 7k\).

Step 6: Finalize the Simplified Expression

The combined fraction will be \(\frac{5k^2 + 7k}{(k + 1)(k + 2)(k + 3)}\). This is the simplified form of the sum of the original fractions.

Key Concepts

Factoring Quadratics

Factoring quadratics is like trying to find what two numbers multiply to make another number. It's a bit like reverse multiplication but with equations. When you see a quadratic, such as in the expressions for our denominators \(k^2 + 4k + 3\) and \(k^2 + 5k + 6\), your job is to break it down.
Imagine you're unwrapping a present; this is what's happening when you "factor" it. Each quadratic can be expressed as a multiplication of two simpler expressions. For instance:

• For \(k^2 + 4k + 3\), we get \((k + 1)(k + 3)\).
• For \(k^2 + 5k + 6\), it's \((k + 2)(k + 3)\).

Knowing these factored forms is essential when you're working with algebraic fractions, especially when you need everything to be neat and organized, like when finding a common denominator.

Common Denominator

A common denominator is like a common meeting point for fractions, so you can add or subtract them easily. Think of two friends who need to meet at the same place. It simplifies everything.
With algebraic fractions, you find a common denominator by determining the least common multiple of the denominators. In our exercise, after factoring the denominators as \((k + 1)(k + 3)\) and \((k + 2)(k + 3)\), we look for the simplest expression that includes all factors from both. This leads us to the common denominator \((k + 1)(k + 2)(k + 3)\).
Finding this helps us combine the fractions accurately, preparing them for addition or subtraction, like getting everyone on the same train heading towards the solution.

Simplifying Expressions

Simplifying expressions is all about making things as clean and straightforward as possible. It's like tidying up a messy room—we want fewer terms and clutter.
In the exercise, once the fractions are rewritten with a common denominator, you'll combine the numerators: \(2k(k + 2) + 3k(k + 1)\). This step involves expanding and simplifying:

• Expand \(2k(k + 2)\) to \(2k^2 + 4k\) and \(3k(k + 1)\) to \(3k^2 + 3k\).
• Combine like terms to end with \(5k^2 + 7k\).

By following these simple steps, you reduce the complexity of the work to a neat expression: \(\frac{5k^2 + 7k}{(k + 1)(k + 2)(k + 3)}\). It makes the previously clunky fractions easier to understand and use.