Problem 15 - Algebra 2

Question

Simplify each sum. \(\frac{5 x}{x^{2}-x-6}+\frac{4}{x^{2}+4 x+4}\)

Step-by-Step Solution

Answer

\(\frac{5 x^{2}+10 x+4}{(x-3)(x+2)^{2}}\).

Step 1: Factorize the denominators

The first fraction has a denominator \(x^{2}-x-6\), which can be factorized into \((x-3)(x+2)\). The second fraction has a denominator \(x^{2}+4x+4\), which can be factored into \((x+2)^{2}\). Thus, the fractions become: \(\frac{5 x}{(x-3)(x+2)}+\frac{4}{(x+2)^{2}}\).

Step 2: Find a common denominator and simplify

The common denominator of the fractions is \((x-3)(x+2)^{2}\). So, we need to multiply the first fraction by \((x+2)/(x+2)\) to get the same denominator for both fractions: \(\frac{5 x(x+2)}{(x-3)(x+2)^{2}}+\frac{4}{(x+2)^{2}}\). After addition and simplification, the result is: \(\frac{5 x^{2}+10 x+4}{(x-3)(x+2)^{2}}\).

Step 3: Combine similar terms

Upon combining similar terms, we get: \(\frac{5 x^{2}+10 x+4}{(x-3)(x+2)^{2}}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of a polynomial. Imagine a polynomial as a huge block tower. Factoring means breaking it down into simpler, smaller blocks.
For example, the polynomial \(x^2 - x - 6\) can be broken into \((x - 3)(x + 2)\). This tells us that if you multiply \((x - 3)\) and \((x + 2)\), you will get back to the original big tower, \(x^2 - x - 6\).

• It makes complex algebraic expressions easier to manage.
• Useful for finding zeros or solving equations.

To factor a polynomial:

• First, try to find two numbers that multiply to the constant term and add to the middle coefficient.
• Then express the polynomial as a product of simpler binomials.

Understanding factoring is like understanding the ingredients in a recipe; it reveals the essential components.

Least Common Denominator

When you want to add or subtract fractions, they need to have the same denominator, just like making sure everyone speaks the same language for better communication. The least common denominator (LCD) is the smallest possible denominator that makes both fractions compatible.
Take two fractions, \(\frac{5x}{(x-3)(x+2)}\) and \(\frac{4}{(x+2)^2}\). They need a common ground, which in this case is \((x-3)(x+2)^2\).

• Multiply the denominators of both fractions to find a common one.
• Adjust the numerators accordingly, being fair and logical as you adjust each fraction to have the same base.

Finding the LCD allows us to simplify the fractions systematically and ensures we can add or subtract them without confusion. Think of it as making sure everyone at the table has equal servings before starting a meal.

Simplifying Expressions

Simplifying expressions is all about tidying up; think of it as cleaning a messy room. The goal is to make expressions neater and easier to understand or work with.
When you simplify, you combine like terms and reduce expressions to their simplest form, removing any clutter or redundancy.

• Using known identities and algebraic rules helps make expressions clearer.
• Always aim to reduce the expression to involve the fewest number of terms with no unnecessary parts.

In our example, after finding the common denominator, the expression \(\frac{5x(x+2) + 4}{(x-3)(x+2)^2}\) becomes \(\frac{5x^2 + 10x + 4}{(x-3)(x+2)^2}\). This form is more streamlined and easier to interpret.
Simplifying is the final touch, much like making a presentation visually appealing to better convey the underlying message.