Problem 10 - Intermediate Algebra

Question

The LCD for \(\frac{2 x+1}{x^{2}+5 x+6}\) and \(\frac{3 x}{x^{2}-4}\) is $$\mathrm{LCD}=(x+2)(x+3)(x-2)$$ If we want to subtract these rational expressions, what form of 1 should be used: a. to build \(\frac{2 x+1}{x^{2}+5 x+6} ?\) b. to build \(\frac{3 x}{x^{2}-4} ?\)

Step-by-Step Solution

Answer

Use \( \frac{x-2}{x-2} \) for \( \frac{2x+1}{x^2+5x+6} \) and \( \frac{x+3}{x+3} \) for \( \frac{3x}{x^2-4} \).

Step 1: Factor the Denominators

First, factor the denominators of the given expressions. For \( \frac{2x+1}{x^2+5x+6} \), the denominator factors into \((x+2)(x+3)\). For \( \frac{3x}{x^2-4} \), the denominator factors into \((x-2)(x+2)\).

Step 2: Identify Missing Factors

Next, identify what factors are missing from each denominator in relation to the LCD. The LCD given is \((x+2)(x+3)(x-2)\). For the first expression, \((x-2)\) is missing. For the second expression, \((x+3)\) is missing.

Step 3: Build the Form of 1 for the First Expression

To rewrite \( \frac{2x+1}{x^2+5x+6} \) with the LCD, multiply the numerator and the denominator by the missing factor \((x-2)\). Thus, the form of 1 to be used is \( \frac{x-2}{x-2} \).

Step 4: Build the Form of 1 for the Second Expression

To rewrite \( \frac{3x}{x^2-4} \) with the LCD, multiply the numerator and the denominator by the missing factor \((x+3)\). Thus, the form of 1 to be used is \( \frac{x+3}{x+3} \).

Key Concepts

Factoring

Factoring is a fundamental process in algebra, crucial for simplifying expressions and solving equations. It involves rewriting an expression as a product of its factors. For instance, the expression \( x^2 + 5x + 6 \) can be factored into \((x+2)(x+3)\), where each factor, when multiplied together, gives the original expression. Similarly, \( x^2 - 4 \) can be factored into \((x-2)(x+2)\) using the difference of squares method.

The key to successful factoring is recognizing special patterns and using them to break down complex expressions:

• **Difference of Squares:** An expression of the form \(a^2 - b^2\) factors into \((a-b)(a+b)\).
• **Perfect Square Trinomials:** Expressions like \(a^2 + 2ab + b^2\) factor into \((a+b)^2\).
• **Common Factoring:** Sometimes the easiest way to factor an expression is by taking out the greatest common factor (GCF) shared among the terms.

Factoring makes working with rational expressions much simpler by breaking down polynomials into more manageable pieces. These simplified expressions are crucial when finding the Least Common Denominator (LCD) or performing operations like addition and subtraction.

Rational Expressions

Rational expressions are fractions where both the numerator and denominator are polynomials. Understanding their properties is essential when dealing with operations like addition, subtraction, multiplication, or division. The key similarity with numeric fractions is the necessity for a common denominator in operations involving addition or subtraction.

To work effectively with rational expressions:

• First, always factor the numerators and denominators where possible. This simplifies the expression and helps identify common factors.
• Identifying the Least Common Denominator (LCD) is crucial. It is the least common multiple of the denominators, ensuring each rational expression can be rewritten with this common base.
• The process of finding an LCD may involve factoring, examining the factors required for each denominator, and then combining them, avoiding duplicates.

Rewriting expressions with the LCD allows them to be easily added or subtracted, as they will share a unified denominator, making other operations straightforward.

Subtracting Fractions

Subtracting fractions, including rational expressions, requires a shared denominator. This necessity stems from ensuring both fractions are "measuring" parts of the same whole, allowing their differences to be logically calculated.

The process can be broken down into key steps:

• **Ensure a Common Denominator:** If the rational expressions don't share a common denominator, identify and use the Least Common Denominator (LCD) to rewrite them equivalently.
• **Rewrite Each Expression:** Multiply the numerator and denominator of each expression by a 'form of 1'—comprised of the factors needed to match the LCD. This doesn't change the value, just its form.
• **Subtract the Numerators:** Once the denominators are the same, subtract the numerators while keeping the denominator constant. It simplifies the operation, focusing on just the numerator adjustments.
• **Simplify the Resulting Expression:** If possible, further simplify the resulting expression by factoring again and reducing it to its simplest form.

Subtracting rational expressions follows the same logical framework as numeric fractions, grounding students in familiar territory while adapting to the nature of polynomials.