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Chapter 7: Problem 63

Divide and, if possible, simplify. $$\frac{25 x^{2}-4}{x^{2}-9} \div \frac{2-5 x}{x+3}$$

Short Answer

Expert verified
\(\frac{-5x - 2}{x - 3}\)

Step by step solution

01

- Rewrite the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. Rewrite the expression as follows:\[\frac{25x^2 - 4}{x^2 - 9} \times \frac{x+3}{2 - 5x}\]
02

- Factor the Numerators and Denominators

Factorize each polynomial in the numerators and denominators where possible.Notice that:1. \(25x^2 - 4\) = \((5x)^2 - 2^2\) = \((5x - 2)(5x + 2)\) (difference of squares).2. \(x^2 - 9\) = \(x^2 - 3^2\) = \((x - 3)(x + 3)\) (difference of squares).3. \(2 - 5x\) can be written as \(-1(5x - 2)\).4. \(x + 3\) remains the same.The expression transforms into:\[\frac{(5x - 2)(5x + 2)}{(x - 3)(x + 3)} \times \frac{x + 3}{-1(5x - 2)}\]
03

- Simplify the Expression

Now, cancel out common factors in the numerators and denominators:1. \((5x - 2)\) cancels out. 2. \((x + 3)\) cancels out.This simplifies the expression to:\[\frac{5x + 2}{-1(x - 3)}\]
04

- Simplify the Final Expression

Simplify \(\frac{5x + 2}{-1(x - 3)}\) to \(\frac{-(5x + 2)}{x - 3}\). Finally, this becomes:\[\frac{-(5x + 2)}{x - 3}\]or \[\frac{-5x - 2}{x - 3}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Division of Fractions
Division of fractions initially seems challenging, but there's a straightforward method to tackle it. When you divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator. For instance, for a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). Applying this to our problem, \(\frac{25 x^2 - 4}{x^2 - 9} \div \frac{2 - 5 x}{x + 3}\) means we rewrite it as \(\frac{25 x^2 - 4}{x^2 - 9} \times \frac{x + 3}{2 - 5 x}\). Now, we've transformed a division problem into a multiplication problem, making it easier to handle.
Factoring Polynomials
Factoring polynomials helps us break down complex expressions into simpler components. It involves writing a polynomial as the product of its factors. In our example, we have the polynomials \(\frac{25 x^2 - 4}{x^2 - 9}\). Both numerators and denominators can be simplified. \(\frac{25 x^2 - 4}{x^2 - 9}\) demonstrates a difference of squares:
  • \(25 x^2 - 4\) becomes \((5x - 2)(5x + 2)\)
  • \(x^2 - 9\) becomes \((x - 3)(x + 3)\)
Additionally, note that \(\frac{2 - 5x}{x + 3}\) can be adjusted to \(\frac{-1(5x - 2)}{x + 3}\) for easier manipulation. This factoring technique is crucial for simplifying rational expressions.
Simplification
Simplification aims to reduce an expression to its simplest form. After factoring, we cancel out common factors in the numerator and denominator. Consider our rewritten expression \(\frac{(5x - 2)(5x + 2)}{(x - 3)(x + 3)} \times \frac{x + 3}{-1(5x - 2)}\). We can observe that both \((5x - 2)\) and \((x + 3)\) are present in the numerator and denominator, allowing us to cancel them out. As a result, the expression simplifies to \(\frac{5x + 2}{-1(x - 3)}\). By simplifying further, we multiply the denominator by -1, obtaining \(\frac{-(5x + 2)}{x - 3}\). Thus, we achieve the simplest possible form.
Reciprocal
The concept of the reciprocal is essential in the division of fractions. It's critical to understand that the reciprocal of a number \(\frac{a}{b}\) is \(\frac{b}{a}\). This principle transforms a division problem into a multiplication problem, making it much simpler to work with. In our exercise, the reciprocal of \(\frac{2 - 5 x}{x + 3}\) is \(\frac{x + 3}{2 - 5 x}\). Factoring further reveals \(\frac{-1(5x - 2)}{x + 3}\). By swapping the numerator and denominator, we convert division into multiplication and apply the reciprocal. Mastering this concept will enhance your efficiency in handling fraction operations.

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