Problem 42 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Example 8. $$ \frac{3 y}{3-x} \div \frac{12 x y}{x^{2}-9} $$

Step-by-Step Solution

Answer

\(-\frac{1}{4x}\)

Step 1: Understanding Division in Fractions

When dividing fractions, the operation can be rewritten as multiplication by the reciprocal of the second fraction. Here, given the expression \( \frac{3 y}{3-x} \div \frac{12 x y}{x^{2}-9} \), we change it to \( \frac{3 y}{3-x} \times \frac{x^{2} - 9}{12 x y} \).

Step 2: Factoring the Denominators

Before proceeding, factor the denominators to simplify the expression. Notice that \( x^{2} - 9 \) is a difference of squares which factors as \( (x-3)(x+3) \). Thus, the expression becomes \( \frac{3 y}{3-x} \times \frac{(x-3)(x+3)}{12 x y} \).

Step 3: Simplifying the Expression

Now, simplify the expression by canceling out common factors in the numerators and denominators. The \( 3y \) in the numerator and denominator cancels, leaving \( \frac{1}{3-x} \times \frac{(x-3)(x+3)}{4x} \). Recognizing that \( 3-x \) can be written as \( -(x-3) \), the expression becomes \( -1 \times \frac{1}{x+3} \times \frac{(x+3)}{4x} \).

Step 4: Final Cancellation

Cancel out \( x+3 \) in the numerator and denominator to further simplify the multiplication. The expression simplifies to \( -\frac{1}{4x} \). Thus, the final answer is \( -\frac{1}{4x} \).

Key Concepts

Multiplying Fractions

Fraction division is closely related to the operation of multiplying fractions. When we divide fractions, we transform the division into multiplication by flipping the second fraction, which involves using its reciprocal. For example, to divide \( \frac{a}{b} \div \frac{c}{d} \), you start by rewriting it as \( \frac{a}{b} \times \frac{d}{c} \). This transformation makes it easier to handle the complex expressions involved.

• Identify the two fractions involved in the division.
• Change the division sign to a multiplication sign.
• Find the reciprocal of the second fraction by swapping its numerator and denominator.
• Proceed with the multiplication of fractions by multiplying the numerators together and the denominators together.

This method streamlines the process and brings consistency to solving problems involving fractions. In our example, turning the division problem into a multiplication problem helped us in managing and simplifying the given expression.

Factoring Techniques

The skill of factoring is crucial when dealing with algebraic expressions, especially for simplifying them. By factoring an expression, we break it down into simpler components that are easier to manipulate and simplify.
To factor effectively:

• Identify any common factors shared by terms, which can be factored out.
• Look for groupings like a difference of squares or trinomials that fit special factoring formulas.
• In our example, recognize that \( x^2 - 9 \) is a difference of squares and can be factorized as \((x-3)(x+3)\).

Employing these factoring techniques aids in transforming complex fractions into manageable pieces, allowing easier simplification when multiplying or dividing fractions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form without changing their value. This usually means canceling or combining like terms, or removing common factors.
Here's how you simplify an expression:

• Reduce expressions by canceling common factors in the numerator and denominator.
• Use associative and commutative properties to rearrange terms, making it simpler to identify like terms.
• In our scenario, cancel \( 3y \) found in both numerator and denominator, simplifying the expression significantly.

Watch out for terms that might look different but are equivalent (such as \( 3-x \) and \(-(x-3)\)), as knowing this can allow further simplification. This can drastically change and often simplify the procedure for obtaining the final, most simple form of an algebraic expression.

Difference of Squares

Recognizing the pattern of a difference of squares is vital in algebra. A difference of squares is a special expression that can always be factored into two conjugate binomials.
The general pattern follows the format \( a^2 - b^2 = (a-b)(a+b) \). This pattern helps in:

• Early identification of expressions that can be easily factored.
• Simplification of complex expressions, especially in fractions.
• Resolving expressions such as \( x^2 - 9 \), where \( 9 \) can be rewritten as \( 3^2 \), leading to the factorization \((x-3)(x+3)\).

Using this technique allows for powerful insights and simplifications, especially when dealing with algebraic expressions embedded in fractions, as it quickly uncovers hidden simplifications, paving the way to a simpler final result.