Problem 47 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{2 x+1}{x^{2}+4 x+4}-\frac{6 x}{x^{2}-4}+\frac{3}{x-2}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-x^2 + x + 10}{(x-2)(x+2)^2} \).

Step 1: Identify the Denominators

The expression is composed of three fractions: \( \frac{2x+1}{x^2 + 4x + 4} \), \( \frac{6x}{x^2 - 4} \), and \( \frac{3}{x-2} \). We start by simplifying the denominators: \( x^2 + 4x + 4 \) can be factored to \((x+2)^2\), \( x^2 - 4 \) is a difference of squares and can be factored to \((x-2)(x+2)\), and the denominator \( x-2 \) is already factored.

Step 2: Find a Common Denominator

The denominators are \((x+2)^2\), \((x-2)(x+2)\), and \((x-2)\). The common denominator will be \((x-2)(x+2)^2\).

Step 3: Rewrite Fractions with Common Denominator

Rewrite each term with the common denominator:- \( \frac{2x+1}{(x+2)^2}\) becomes \( \frac{2x+1}{(x+2)^2} \times \frac{x-2}{x-2} = \frac{(2x+1)(x-2)}{(x-2)(x+2)^2} \)- \( \frac{6x}{(x-2)(x+2)}\) becomes \( \frac{6x}{(x-2)(x+2)} \times \frac{x+2}{x+2} = \frac{6x(x+2)}{(x-2)(x+2)^2} \)- \( \frac{3}{x-2}\) becomes \( \frac{3}{x-2} \times \frac{(x+2)^2}{(x+2)^2} = \frac{3(x+2)^2}{(x-2)(x+2)^2} \)

Step 4: Combine the Numerators

Combine the rewritten fractions to have a single fraction with the denominator \((x-2)(x+2)^2\). Combine the numerators:\[ (2x+1)(x-2) - 6x(x+2) + 3(x+2)^2 \]

Step 5: Expand and Simplify the Numerator

Expand each term in the numerator:- \((2x+1)(x-2)\) expands to \(2x^2 - 4x + x - 2 = 2x^2 - 3x - 2\)- \(-6x(x+2)\) expands to \(-6x^2 - 12x\)- \(3(x+2)^2\) expands to \(3(x^2 + 4x + 4) = 3x^2 + 12x + 12\)Combine the terms: \[ 2x^2 - 3x - 2 - 6x^2 - 12x + 3x^2 + 12x + 12 = -x^2 + (x + 12) - 2 \]

Step 6: Final Simplification

Simplify the combined terms:\[-x^2 + x + 10\] which makes the simplified numerator.So the final expression becomes:\[ \frac{-x^2 + x + 10}{(x-2)(x+2)^2} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is crucial when dealing with rational expressions. It involves breaking down a polynomial into simpler, multiplicative components, which can reveal common factors with other expressions. For instance, in our exercise, the polynomial \(x^2 + 4x + 4\) is factored into \((x+2)^2\). This is a perfect square trinomial, which results in the square of a binomial.

Similarly, \(x^2 - 4\) is a difference of squares. The formula for the difference of squares \(a^2 - b^2\) is \((a+b)(a-b)\). Applying this, we factor \(x^2 - 4\) into \((x-2)(x+2)\). These factored forms are easier to work with when simplifying or finding common denominators in rational expressions.

The main benefits of factoring include simplifying expressions, aiding in finding common denominators, and facilitating the simplification of complex rational expressions. Remember, identifying and applying the right factorization technique is key to success in algebra.

Common Denominator

Finding a common denominator is essential when you add or subtract fractions, especially with different denominators. A common denominator allows you to combine fractions by rewriting each fraction so that they share the same bottom part. The goal is to find the least common multiple of all the denominators involved.

In the exercise, we have the denominators \((x+2)^2\), \((x-2)(x+2)\), and \(x-2\). The common denominator that encompasses all components is \((x-2)(x+2)^2\). This ensures that every term can be converted into a fraction with this denominator.

The procedure involves multiplying each fraction by a form of 'one' (something that equals the common denominator divided by the original denominator). This transformation allows the numerators to be combined directly, as they now share a unified base. It is an essential step in rational expressions that ensures accurate addition or subtraction of fractions.

Simplifying Fractions

Simplifying fractions in rational expressions involves combining and reducing them to their simplest form. This often requires expanding and combining terms, followed by cancelling out common factors from the numerator and the denominator wherever possible.

In the step-by-step solution, the numerators of each term are first expanded separately: \((2x+1)(x-2)\) expands to \(2x^2 - 3x - 2\), \(-6x(x+2)\) to \(-6x^2 - 12x\), and \(3(x+2)^2\) to \(3x^2 + 12x + 12\). These expansions are combined to form a single expression. Combining like terms results in the simplified final numerator \(-x^2 + x + 10\).

Once combined, the simplification may not always result in reducing the numerator and denominator, but it does contest the expression to its simplest operational form. Simplifying ensures that the rational expression is as efficient and readable as possible. It is a vital step to help clarify complex fractions and makes further calculations much easier.