Problem 41 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{x^{2}+5 x+6}{3 x y}}{\frac{9-x^{2}}{6 x y}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{-2(x+2)}{x-3} \).

Step 1: Identify the Complex Fraction

The complex fraction given is \( \frac{\frac{x^{2}+5x+6}{3xy}}{\frac{9-x^{2}}{6xy}} \). We have a fraction on top and another on the bottom.

Step 2: Simplify the Numerator and Denominator

First, simplify the numerator \( x^2 + 5x + 6 \) and the denominator \( 9 - x^2 \) by factoring.- For \( x^2 + 5x + 6 \), it factors to \((x+2)(x+3)\).- For \( 9 - x^2 \), it can be rewritten as \((3-x)(3+x)\) or \(-(x-3)(x+3)\).

Step 3: Rewrite the Complex Fraction

After factoring, the complex fraction becomes: \[ \frac{\frac{(x+2)(x+3)}{3xy}}{\frac{-(x-3)(x+3)}{6xy}} \].

Step 4: Simplify by Multiplying by the Reciprocal

Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator:\[ \frac{(x+2)(x+3)}{3xy} \times \frac{6xy}{-(x-3)(x+3)} \].

Step 5: Cancel Common Terms

Cancel out the common terms in the numerator and the denominator:- Cancel \( x+3 \) and \( xy \) from both the numerator and denominator.- Simplify to \(-2 \times \frac{x+2}{x-3}\) after cancellations.

Step 6: Write the Final Simplified Form

The simplified form of the complex fraction is: \[ \frac{-2(x+2)}{x-3} \].

Key Concepts

Factoring Polynomials

Factoring polynomials is akin to breaking them down into simpler building blocks. This concept is essential when working with complex fractions because it makes the simplification process manageable. When you factor a polynomial, you're looking for two or more expressions that multiply together to give you the original polynomial. For example, with the polynomial \(x^2 + 5x + 6\), the factors are \((x+2)(x+3)\), because \((x+2)\times(x+3) = x^2 + 5x + 6\).

During factoring, it's crucial to identify common patterns such as the difference of squares, which helps in rewriting expressions more efficiently. In our exercise, the denominator \(9 - x^2\) is a difference of squares and factors into \((3-x)(3+x)\) or equivalently \(-(x-3)(x+3)\).

• Look for integer factors that sum to the middle coefficient.
• Apply special formulas like the difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator contain polynomials. In our exercise, both the numerator \(\frac{x^2+5x+6}{3xy}\) and the denominator \(\frac{9-x^2}{6xy}\) are rational expressions. Understanding these expressions requires knowing how to factor and simplify them.

The key steps in managing rational expressions include:

• Factoring the polynomials in both the numerator and the denominator.
• Simplifying the fraction by canceling out common factors.
• Rewriting division as multiplication by the reciprocal in complex fractions.

By handling rational expressions properly, you can ease the process of simplifying complex fractions. In our example, the simplification involves rewriting and simplifying the expression before multiplying by the reciprocal.

Simplification Techniques

Simplifying complex fractions involves several techniques that help break down larger expressions into simpler ones. The process generally includes the identification of the fraction structure, factoring polynomials, and multiplying by the reciprocal.

In our solved exercise, we start by identifying the structure \(\frac{\text{Fraction 1}}{\text{Fraction 2}}\), which necessitates factoring polynomials in each part of the fraction. Once the polynomials are factored, rewriting the division of fractions as a multiplication by the reciprocal simplifies the complex fraction:

• Convert \(\frac{\text{Fraction 1}}{\text{Fraction 2}}\) into \(\text{Fraction 1} \times \frac{1}{\text{Fraction 2}}\).
• Cancel common factors to avoid unnecessary complications in the expression.
• Simplify what's left to get to the lowest terms.

By applying these techniques, the original complex fraction simplifies from a potentially messy expression to \(\frac{-2(x+2)}{x-3}\), showing the power of careful and methodical simplification.