Problem 24 - Algebra A Combined Function

Question

Find each quotient and simplify. See Examples 4 through 7. $$ \frac{9 x^{5}}{a^{2}-b^{2}} \div \frac{27 x^{2}}{3 b-3 a} $$

Step-by-Step Solution

Answer

The simplified quotient is \(-\frac{x^3}{a+b}\).

Step 1: Rewrite the division as multiplication by reciprocal

The division of two fractions is equivalent to the multiplication of the first fraction by the reciprocal of the second fraction. Thus, you can rewrite the expression as follows:\[\frac{9x^5}{a^{2}-b^{2}} \div \frac{27x^2}{3b-3a} = \frac{9x^5}{a^{2}-b^{2}} \times \frac{3b-3a}{27x^2}\]

Step 2: Simplify the expression

Now, simplify each part of the expression:1. **Denominators simplification:** Recognize that \(a^{2}-b^{2}\) is a difference of squares, which factors to \((a-b)(a+b)\).2. **Numerators simplification:** Recognize 3b - 3a can be factored as \(-3(a-b)\). Substituting the fraction with these factors, you find:\[\frac{9x^5}{(a-b)(a+b)} \times \frac{-3(a-b)}{27x^2}\]

Step 3: Multiply the fractions and simplify

Multiply the numerators together and the denominators together:\[\frac{9x^5 \times -3(a-b)}{(a-b)(a+b) \times 27x^2}\]Cancel common factors present in the numerator and the denominator. For example, \(9x^5\) and \(27x^2\) can be reduced by their greatest common divisor, which is \(9x^2\), leading to \(x^3\) over 3. Also, \((a-b)\) terms can be canceled from both the numerator and the denominator:\[\frac{-1 \times 3x^3}{(a+b) \times 3}\]

Step 4: Final simplification

Simplify the fraction further:\[\frac{-3x^3}{3(a+b)} = \frac{-x^3}{a+b}\]The quotient is simplified to:\[-\frac{x^3}{a+b}\]

Key Concepts

Difference of Squares

The **Difference of Squares** is a fundamental algebraic concept often used to simplify expressions. It involves identifying and factoring expressions of the form \(a^{2} - b^{2}\). This expression can be rewritten as \((a + b)(a - b)\). Recognizing this pattern allows us to break down polynomials into simpler binomials.

• If you see a quadratic term minus another quadratic term, think about the difference of squares.
• This technique is particularly useful in simplifying algebraic fractions where these patterns emerge.

In our exercise, the term \(a^{2} - b^{2}\) was factored as \((a - b)(a + b)\), providing a simpler form that could be used in the subsequent steps. Such transformations make operations like multiplication or division clearer and more manageable.

Reciprocal of a Fraction

The **Reciprocal of a Fraction** is crucial for transforming division into multiplication, a fundamental step in simplifying algebraic expressions. Every fraction \(\frac{a}{b}\) has a reciprocal \(\frac{b}{a}\).

• To divide by a fraction, you multiply by its reciprocal. This is because multiplying by the reciprocal reverses the division.
• This technique is essential to ease calculations, especially when handling complex algebraic terms.

By converting the division in our exercise into multiplication by the reciprocal, the problem becomes more straightforward to solve. This strategy helps maintain the accuracy of operations, especially with variables involved.

Fraction Multiplication

**Fraction Multiplication** involves multiplying the numerators together and the denominators together to find the product. This process may seem straightforward, but the key lies in **simplifying** before and after multiplying.

• Always factor expressions before multiplying to identify and cancel out any common factors, which simplifies the fraction efficiently.
• Multiplication often involves handling large polynomials; reducing complexity through simplification and cancellation of terms can save a lot of time.

In the provided solution, the fractions were multiplied together after transforming the division problem using the reciprocal. This step included canceling out common factors like \(9x^2\) and \((a-b)\), streamlining the process and preventing errors.

Factoring Polynomials

**Factoring Polynomials** is the process of breaking down a polynomial into a product of simpler polynomials. This is a crucial skill for simplifying algebraic expressions and is often employed in simplifying fractions or solving polynomial equations.

• Recognize patterns such as the **difference of squares** or **common factors** that can be factored from terms.
• Factorization makes complex algebraic operations like multiplication and division simpler and more efficient.

In our example, factoring was used to simplify both the numerator \(3b - 3a\), factorable to \(-3(a-b)\), and the denominator \(a^2 - b^2\), factorable to \((a-b)(a+b)\). Efficient use of factoring reduces the fractions to simpler forms, allowing further operations without unnecessary complexity.