Problem 23 - Algebra A Combined Function

Question

Find each quotient and simplify. See Examples 4 through 7. $$ \frac{3 x^{2}}{x^{2}-1} \div \frac{x^{5}}{(x+1)^{2}} $$

Step-by-Step Solution

Answer

\( \frac{3(x+1)}{x^3(x-1)} \)

Step 1: Rewrite the Division as Multiplication

To simplify the expression \( \frac{3x^2}{x^2-1} \div \frac{x^5}{(x+1)^2} \), we start by rewriting the division as multiplication by flipping the second fraction. So, \( \frac{3x^2}{x^2-1} \div \frac{x^5}{(x+1)^2} = \frac{3x^2}{x^2-1} \times \frac{(x+1)^2}{x^5} \).

Step 2: Factor the Denominator

Recognize that the denominator of the first fraction, \( x^2 - 1 \), is a difference of squares: \( x^2 - 1 = (x+1)(x-1) \).So, \( \frac{3x^2}{x^2-1} = \frac{3x^2}{(x+1)(x-1)} \).

Step 3: Multiply the Fractions

Now multiply the numerators together and the denominators together:\[ \frac{3x^2 \cdot (x+1)^2}{(x+1)(x-1) \cdot x^5} \].

Step 4: Simplify the Expression

Cancel out the common factor \( (x+1) \) in the numerator and denominator:\[ \frac{3x^2 \cdot (x+1)}{(x-1) \cdot x^5} \].Further, simplify \( x^2 \) in the numerator with \( x^5 \) in the denominator:\[ \frac{3(x+1)}{(x-1)x^3} \].

Step 5: Final Simplified Form

The simplified form of the expression is \( \frac{3(x+1)}{x^3(x-1)} \). No further simplifications are possible.

Key Concepts

Division of Fractions

When dividing fractions, it's helpful to understand that you're essentially performing multiplication by the reciprocal. In basic terms, this means flipping the second fraction, which involves swapping its numerator and denominator, and then changing the operation from division to multiplication. For example, dividing by the fraction \( \frac{a}{b} \) is the same as multiplying by \( \frac{b}{a} \).
This method transforms a potentially complex division problem into a more manageable multiplication problem. It can simplify your calculations and makes it easier to see where factors can be canceled out.
So, for the given problem, we perform this step by rewriting \( \frac{3x^2}{x^2-1} \div \frac{x^5}{(x+1)^2} \) as \( \frac{3x^2}{x^2-1} \times \frac{(x+1)^2}{x^5} \), which sets us up for easier simplification in the next steps.

Simplification

Simplification involves reducing the expression to its simplest form while retaining equivalence. The simplification process helps in identifying common factors in both the numerator and denominator, which allows for canceling them out.
In our expression, we first multiply the numerators and the denominators. This gives\[ \frac{3x^2 \cdot (x+1)^2}{(x+1)(x-1) \cdot x^5} \].

• The common factors, such as \((x+1)\) in the numerator and denominator, can be canceled.
• The factors \(x^2\) and \(x^5\) are simplified by subtracting exponents, resulting in \(x^3\) in the denominator.

This step-by-step cancellation reduces clutter in the expression and provides a clearer view of the simplest equivalent form, which is \( \frac{3(x+1)}{x^3(x-1)} \). This technique is vital in algebra as it makes further mathematical operations more straightforward and less prone to errors.

Factoring Polynomials

Factoring polynomials is a powerful technique that can greatly aid in simplification processes like dividing algebraic fractions. Factoring involves breaking down a complex expression into simpler multiplicative components. This strategy can highlight common factors readily, making cancelation quicker and easier.
In the original exercise, the expression \( x^2 - 1 \) is identified as a difference of squares. This specific type of polynomial can be factored as \((x+1)(x-1)\). Recognizing and leveraging such patterns is vital:

• Difference of squares: \( a^2 - b^2 = (a-b)(a+b) \)
• Perfect square trinomials: \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \)

Once factored, identifying and canceling common factors becomes straightforward. In our given expression, this step leads to a more manageable problem and brings clarity, easing subsequent calculations.