Problem 44 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{5 x}{2 x+3}-\frac{6}{2 x^{2}+3 x}+\frac{2}{x}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5x + 4}{2x + 3} \).

Step 1: Identify the common denominator

We have three fractions: \( \frac{5x}{2x+3} \), \( \frac{6}{2x^2+3x} \), and \( \frac{2}{x} \).To simplify, we need a common denominator. First, factor the second denominator: \( 2x^2 + 3x = x(2x+3) \). Thus, the common denominator for these fractions is \( x(2x+3) \).

Step 2: Re-write each fraction with the common denominator

Convert each fraction to have the common denominator \( x(2x+3) \):- \( \frac{5x}{2x+3} = \frac{5x \cdot x}{x(2x+3)} = \frac{5x^2}{x(2x+3)} \)- \( \frac{6}{2x^2+3x} = \frac{6}{x(2x+3)} \) (already in desired form)- \( \frac{2}{x} = \frac{2 \cdot (2x+3)}{x(2x+3)} = \frac{4x+6}{x(2x+3)} \)

Step 3: Combine the fractions

Add and subtract the new expressions that have the common denominator: \[ \frac{5x^2}{x(2x+3)} - \frac{6}{x(2x+3)} + \frac{4x+6}{x(2x+3)} = \frac{5x^2 - 6 + 4x + 6}{x(2x+3)} \] Combine the terms in the numerator.

Step 4: Simplify the expression

Simplify the expression in the numerator: \[ 5x^2 + 4x + 6 - 6 = 5x^2 + 4x \]The simplified expression becomes: \[ \frac{5x^2 + 4x}{x(2x+3)} \]Check if further simplification is possible.

Step 5: Simplify if possible

Factor out the common term \(x\) in the numerator \(5x^2 + 4x = x(5x + 4)\). The expression is now \[ \frac{x(5x+4)}{x(2x+3)} \].Cancel out the \(x\) in the numerator and denominator, assuming \(x eq 0\).

Step 6: Final simplified expression

After canceling the common \(x\) term, the expression becomes: \[ \frac{5x + 4}{2x + 3} \].This is the simplified form of the given expression.

Key Concepts

Common Denominator

When simplifying fractional expressions, finding a common denominator is crucial. The denominator is the number at the bottom of a fraction, representing the total parts of a whole. Here we have three fractions: \( \frac{5x}{2x+3} \), \( \frac{6}{2x^2+3x} \), and \( \frac{2}{x} \). To combine these fractions, they need the same denominator. This is similar to finding a common language.Here’s how to find it:

• Factor the denominators. For instance, \( 2x^2 + 3x \) can be factored to \( x(2x+3) \).
• From all the factored denominators, the common denominator becomes \( x(2x+3) \).

With the common denominator \( x(2x+3) \), you can easily rewrite each fraction and perform operations like addition or subtraction. This makes the simplification process much smoother.

Factorization

Factorization is breaking down an expression into multiples that can be multiplied together to get back the original expression. Imagine taking a complicated expression and breaking it into simple building blocks.In our exercise, factorization helps us simplify denominators, especially for terms like \( 2x^2 + 3x \). You can factor this using the following method:

• Common factor extraction. Both terms have an \( x \). Extract it: \( x(2x+3) \).

Recognizing these patterns in factorization makes identifying a common denominator straightforward. It simplifies the algebraic expression, making it easier to combine multiple fractions. Remember, factorization is like finding common threads in different fabrics of the algebraic world.

Fractional Expressions

A fractional expression is an algebraic expression containing one or more fractions. They can be tricky to work with unless simplified correctly. Consider fractional expressions as mini-equations within a larger equation, with numerators and denominators involving variables and constants.Here is why they are important:

• They are used to express divisions of quantities.
• Simplifying them requires finding common denominators and applying operations like additions, subtractions, or multiplications.

In our task, each fraction is expressed initially with different denominators, making their direct combination impossible. By rewriting each fraction over a common denominator, \( x(2x+3) \), it becomes easier to treat all fractions as parts of one big family, thus simplifying the overall expression.

Simplification Steps

Simplification involves reducing an expression to its simplest form, stripping away any complexities without changing its value. In algebra, this process often involves a series of steps. Here’s a clear path to simplification:

• Identify a common denominator, then rewrite each fraction to have this denominator.
• Combine the numerators into a single expression, as they all now share the same base.
• Simplify the combined expression by performing operations like addition, subtraction, or factoring.

In our example, after rewriting all fractions with a common denominator \( x(2x+3) \), we combined and simplified numerators by collecting like terms and factoring when possible. Eventually, canceling common terms in the numerator and denominator leads to the final simplified expression \( \frac{5x + 4}{2x + 3} \). This ensures the expression is as simple as can be for further mathematical tasks.