Problem 50 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{x^{2}-9}{5 x+15} \div \frac{3-x}{x+3}\)

Step-by-Step Solution

Answer

The simplified result is \(-\frac{1}{5}\).

Step 1: Rewrite Division as Multiplication

When dividing fractions, rewrite the division problem as multiplication by the reciprocal of the divisor. For the given expression, change \( \frac{x^{2}-9}{5x+15} \div \frac{3-x}{x+3} \) to \( \frac{x^{2}-9}{5x+15} \times \frac{x+3}{3-x} \).

Step 2: Factor the Numerators and Denominators

Identify the factorable components in each part of the expression. Notice that \(x^2 - 9\) is a difference of squares: \(x^2 - 9 = (x-3)(x+3)\). Similarly, factor \(5x + 15\) as \(5(x+3)\). The expression becomes \(\frac{(x-3)(x+3)}{5(x+3)} \times \frac{x+3}{3-x}\).

Step 3: Simplify the Expression

Cancel out the common terms in the numerator and the denominator. The \(x+3\) term appears in both the numerator and denominator, so cancel them out. This simplifies the expression to \(\frac{x-3}{5} \times \frac{1}{3-x}\).

Step 4: Simplify the Multiplication Result

Recognize that \(3-x\) can be expressed as \(-(x-3)\). Therefore, \(\frac{1}{3-x} = \frac{-1}{x-3}\). Substitute this into the equation: \(\frac{x-3}{5} \times \frac{-1}{x-3} = \frac{-(x-3)}{5(x-3)}\).

Step 5: Cancel and Finalize the Simplified Result

Cancel the \(x-3\) term in both the numerator and denominator, noting that the final expression becomes \(\frac{-1}{5}\). This is the simplified version of the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential step when working with rational expressions, as it helps simplify the expressions and make calculations manageable. The main goal is to break down a complex expression into simpler components, often called factors.
For example, in the given expression, we encountered the polynomial \(x^2 - 9\). This can be factored as a difference of squares: \(x^2 - 9 = (x-3)(x+3)\).

• A difference of squares follows the format \(a^2 - b^2 = (a-b)(a+b)\).
• This technique is applicable to any polynomial that fits this pattern.

By factoring correctly, you can reveal hidden simplifications and work towards a simplified solution.
Another component in our exercise, \(5x + 15\), can be factored by extracting the greatest common factor (GCF), which is 5 in this case. So, it becomes \(5(x+3)\).
Understanding polynomial factoring is like finding smaller building blocks that construct the original expression. The better you can or identify these factors, the simpler it is to handle larger expressions.

Dividing Fractions

Dividing fractions may seem daunting at first, but there's a straightforward method that can make the process easy to handle. The key concept here is to switch the division operation to multiplication. To do this, you multiply by the reciprocal of the divisor.
Consider the expression \(\frac{x^2-9}{5x+15} \div \frac{3-x}{x+3}\). Instead of directly dividing, we flip the second fraction and multiply it: \(\frac{x^2-9}{5x+15} \times \frac{x+3}{3-x}\).

• The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
• This transformation allows you to use easier multiplication rules instead of the more complex division.

This switch is made possible by understanding that multiplying by a reciprocal is equivalent to dividing, but more intuitive to compute.
Bearing this method in mind can turn challenging problems into simpler calculations, ensuring you can handle any fraction division easily.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, often by canceling out common factors or terms. This process helps to present expressions in a more understandable and usable format.
For instance, in our problem after multiplication, we get \(\frac{(x-3)(x+3)}{5(x+3)} \times \frac{x+3}{3-x}\).

• First, identify and cancel common elements in the numerator and denominator across the fractions.
• Here, the term \(x+3\) in both fractions cancels out.
• Later, simplify \(3-x\) to \(-(x-3)\). This means \(\frac{1}{3-x} = \frac{-1}{x-3}\).

Continuing simplification, multiply the terms: \(\frac{x-3}{5} \times \frac{-1}{x-3}\), allowing the \(x-3\) to cancel out entirely, bringing us to the final answer, \(\frac{-1}{5}\).
Simplifying expressions lets you see the bigger picture without distractions from unnecessary complexity.