Problem 123 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\left(\frac{3}{x-3}-\frac{1}{x}\right) \div \frac{12 x+18}{x^{3}-9 x}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{x+3}{6}\).

Step 1: Express Division as Multiplication

The given expression involves division. To simplify, let's convert the division into multiplication by taking the reciprocal of the second fraction:\[\left(\frac{3}{x-3}-\frac{1}{x}\right) \times \frac{x^3 - 9x}{12x + 18}\]

Step 2: Factor the Denominator and Numerator

Let's factor the expressions where possible. First, we'll factor the terms:- In the denominator of the second fraction, \(12x + 18\) factors to \(6(2x + 3)\).- The numerator \(x^3 - 9x\) can be factored as \(x(x^2 - 9) = x(x - 3)(x + 3)\).Thus, the expression becomes:\[\left(\frac{3}{x-3}-\frac{1}{x}\right) \times \frac{x(x-3)(x+3)}{6(2x+3)}\]

Step 3: Find a Common Denominator for the Fraction Subtraction

To subtract the fractions in the first part, \(\frac{3}{x-3}-\frac{1}{x}\), we need a common denominator, which will be \((x-3)x\). Rewriting the fractions we get:- \(\frac{3}{x-3} = \frac{3x}{x(x-3)}\)- \(\frac{1}{x} = \frac{x-3}{x(x-3)}\)Subtracting these, the expression becomes:\[\frac{3x - (x-3)}{x(x-3)} = \frac{3x - x + 3}{x(x-3)} = \frac{2x + 3}{x(x-3)}\]

Step 4: Multiply the Expressions

Now multiply the simplified expression: \[\frac{2x + 3}{x(x-3)} \times \frac{x(x-3)(x+3)}{6(2x+3)}\]Cancel out identical terms in the numerator and denominator:- \(x(x-3)\) cancels with \(x(x-3)\).- \(2x+3\) cancels with \(2x+3\).This leaves us with:\[\frac{x+3}{6}\]

Step 5: Simplified Result

After performing all the cancellation, the entire expression simplifies to:\[\frac{x+3}{6}\]This is the most reduced form of the original problem, as no further simplification is possible.

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions that contain polynomials in both the numerator and the denominator. Simplifying rational expressions is essentially about finding their most reduced form.
To simplify, follow these steps:

• Factor both the numerator and the denominator completely. This involves rewriting polynomials as products of their factors.
• Cancel out common factors from the numerator and the denominator.
• Rearrange the expression if needed to avoid any undefined terms, such as division by zero.

A simplified rational expression will have no common factors other than 1.
It's important to remember that while simplifying, the expression should remain equivalent to the original expression.

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions.
It involves expressing a polynomial as a product of its simplest components. These are often called factors.
Here are the types of factorizations you might encounter:

• **Common factor:** Always check for a common factor in all terms of the polynomial.
• **Difference of squares:** Applies to expressions like \(a^2 - b^2\), factored into \((a-b)(a+b)\).
• **Trinomials:** Quadratic expressions can be factored into the product of two binomials.

For example, the polynomial \(x^3 - 9x\) can be factored by first factoring out the common factor \(x\), resulting in \(x(x^2 - 9)\), and then recognizing it as a difference of squares: \(x(x-3)(x+3)\).
Mastering factoring techniques simplifies both solving and simplifying polynomial-based problems.

Fraction Operations

Fraction operations often involve addition, subtraction, multiplication, or division. Each type of operation follows specific rules to maintain the value of the fractions.

For addition and subtraction:

• Identify or create a common denominator between the fractions.
• Re-write each fraction to have this common denominator.
• Perform the addition or subtraction on the numerators only, keeping the common denominator unchanged.

For multiplication:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Simplify the resulting fraction by canceling common factors.

For division:

• Take the reciprocal of the second fraction.
• Change the division problem into a multiplication problem.
• Proceed as you would for multiplication.

Utilizing these methods makes handling fractions straightforward and ensures accuracy in calculations.