Problem 28 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{x^{2}+15 x+54}{x^{2}+2} \cdot \frac{3 x^{2}+6}{x^{2}+10 x+9}$$

Step-by-Step Solution

Answer

\( \frac{3(x+6)}{x+1} \)

Step 1: Factor the Expressions

First, we need to factor the numerators and denominators of both fractions.\(\frac{x^{2}+15 x+54}{x^{2}+2}\) and \(\frac{3 x^{2}+6}{x^{2}+10 x+9}\).1. Factor \(x^{2}+15x+54\) as \((x+6)(x+9)\).2. Factor \(3x^2+6\) as \(3(x^2+2)\).3. Factor \(x^2+10x+9\) as \((x+1)(x+9)\).

Step 2: Substitute Factored Forms

Substitute the factored forms into the original expression: \[\frac{(x+6)(x+9)}{x^2+2} \cdot \frac{3(x^2+2)}{(x+1)(x+9)}\]

Step 3: Cancel Common Factors

Next, cancel out any common factors in the numerators and denominators: The common factors between the two fractions are \(x^2+2\) and \(x+9\).Thus the expression simplifies to: \[ \frac{3(x+6)}{x+1} \]

Step 4: Simplify Further

No further simplification is possible as there are no more common factors. The expression is already in its simplest form.Therefore, the answer is \( \frac{3(x+6)}{x+1} \).

Key Concepts

Simplification of Rational Expressions

To simplify rational expressions, we resemble the process of simplifying numerical fractions. The goal is to make the expression easier to understand and work with by reducing it to its simplest form. This involves:

• Factoring both the numerator and the denominator.
• Cancelling out common factors between them.

By removing these common factors, we streamline the expression and expose any hidden redundancy. This involves a step-by-step approach:

• Identify parts of the expression that can be factored.
• Simplify each part separately before recombining.
• Re-examine the entire expression for additional simplifications.

Consider that you should cease simplification once all common factors are eliminated. This is because further intervention might potentially complicate the expression rather than simplify it further.

Multiplying Rational Expressions

Multiplication of rational expressions involves the same principles as multiplying fractions. You multiply across the numerators and denominators:

• Multiply the numerators together to find the new numerator.
• Multiply the denominators together to find the new denominator.

After performing the multiplication, the crucial step is simplification. In this phase:

• Factor the resulting expression whenever possible.
• Cancel any common factors appearing in both the numerator and the denominator.

By focusing on identifying and canceling these common factors, the resultant expression becomes easier to work with. This not only consolidates the expression but also reduces computational complexity in subsequent operations.

Algebraic Fractions

Algebraic fractions are essentially fractions where both the numerator and the denominator are algebraic expressions. Handling these fractions requires:

• Understanding how to manipulate variables just like numbers.
• Mastering common algebraic techniques such as factoring and expanding.

By interpreting algebraic fractions this way, we can apply arithmetic operations (addition, subtraction, multiplication, division) to them.
Managing algebraic fractions implies:

• Performing operations separately on the numerators and denominators.
• Simplifying where possible to reduce the complexity of algebraic fractions.

Grasping this concept grounds you in algebra and enhances your problem-solving skills, allowing you to tackle more complex operations with ease.