Problem 62 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{6 x-12}{8 x+32} \div \frac{18 x-36}{10 x+40}$$

Step-by-Step Solution

Answer

The result of the division is \(\frac{5}{12}\).

Step 1: Simplify Each Rational Expression

Firstly, you simplify each of the rational expressions:\[\frac{6x-12}{8x+32} \rightarrow \frac{6(x-2)}{8(x+4)} \rightarrow \frac{3(x-2)}{4(x+4)}\]\[\frac{18x-36}{10x+40} \rightarrow \frac{18(x-2)}{10(x+4)} \rightarrow \frac{9(x-2)}{5(x+4)}\]

Step 2: Change Division to Multiplication

The division of fractions can be changed to multiplication by taking the reciprocal of the divisor. So, the division changes to:\[\frac{3(x-2)}{4(x+4)} \times \frac{5(x+4)}{9(x-2)}\]

Step 3: Perform Multiplication

Multiply the numerators together and the denominators together:\[\frac{15(x-2)(x+4)}{36(x-2)(x+4)}\]

Step 4: Simplify the Result

Now, cancel out the common terms in the numerator and the denominator:\[\frac{15(x-2)(x+4)}{36(x-2)(x+4)} \rightarrow \frac{15}{36}\]which simplifies to \[\frac{5}{12}\]

Key Concepts

Simplifying Algebraic Fractions

Simplifying algebraic fractions is a critical skill for anyone studying algebra. To start simplifying an algebraic fraction, look for common factors in the numerator and denominator that you can divide by. This process is similar to simplifying numerical fractions by canceling common numerical factors.

In the given exercise, the expression \(\frac{6x-12}{8x+32}\) simplifies to \(\frac{3(x-2)}{4(x+4)}\) by factoring out 6 in the numerator and 8 in the denominator, leaving \(x-2\) and \(x+4\) as the remaining terms. The essence of simplifying is reducing the complexity of the expression, making it easier to handle in further calculations. Recognizing common factors is essential, as is factoring polynomials - a process we'll dive into in a later section.

Multiplication of Fractions

Multiplication of fractions is quite straightforward: simply multiply the numerators together and the denominators together. This principle applies to both numerical and algebraic fractions.

In our exercise, after we convert the division of the two algebraic fractions into multiplication by taking the reciprocal of the second fraction, \(\frac{18x-36}{10x+40}\), we get \(\frac{3(x-2)}{4(x+4)} \times \frac{5(x+4)}{9(x-2)}\). Here, we multiply the numerators \(3(x-2)\) and \(5(x+4)\), and the denominators \(4(x+4)\) and \(9(x-2)\), to find the product of the fractions.

Reciprocal of a Fraction

The reciprocal of a fraction is obtained by swapping its numerator and denominator. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\), provided that neither \(a\) nor \(b\) is zero (as division by zero is undefined).

The concept of reciprocation is particularly useful when dividing fractions. In the exercise, to divide by the fraction \(\frac{18x-36}{10x+40}\), we take its reciprocal and multiply instead, arriving at a new algebraic fraction which is easier to multiply. Understanding reciprocals is not only key in division but also in understanding mathematical relationships and functions.

Factoring Polynomials

Factoring polynomials is a method of expressing the polynomial as the product of its factors, which are usually simpler algebraic expressions. In the context of simplifying algebraic fractions, factoring allows us to identify and cancel out common algebraic factors.

Let's consider the polynomials in the original exercise. The term \(6x-12\) can be factored as \(6(x-2)\) and \(8x+32\) as \(8(x+4)\). Similarly, \(18x-36\) factors as \(18(x-2)\) and \(10x+40\) as \(10(x+4)\). Factoring is a powerful tool that simplifies both the numerator and the denominator of algebraic fractions, facilitating further operations like multiplication, division, addition, and subtraction.