Problem 82 - Prealgebra and Introductory Algebra

Question

State whether the given division is equivalent to \(\frac{x^{2}-3 x-4}{x^{2}+5 x-6}\). $$\frac{x+1}{x-1} \div \frac{x+6}{x-4}$$

Step-by-Step Solution

Answer

Yes, the given division is equivalent to \(\frac{x^2-3x-4}{x^2+5x-6} \).

Step 1: Simplify Given Fractions

Simplify each fraction in the division expression. This simplification involves factoring the numerator and denominator of each fraction where possible. The fractions become: \( \frac{x+1}{x-1} \) and \( \frac{x+6}{x-4} \)

Step 2: Perform The Division

Perform the division operation on the two simplified expressions. In the case of fractions, this involves flipping (finding the reciprocal of) the second fraction and changing the operation to multiplication. Thus, \(\frac{x+1}{x-1} \div \frac{x+6}{x-4} \) becomes \(\frac{x+1}{x-1} \cdot \frac{x-4}{x+6} \). Calculate and simplify this multiplication to get: \(\frac{x^2-4x+x-4}{(x-1)(x+6)}\) which simplifies further to \(\frac{x^2-3x-4}{x^2+5x-6} \).

Step 3: Compare The Two Expressions

Now that both expressions are simplified, they can be compared. The expression derived from the multiplication that resulted from our division operation is \(\frac{x^2-3x-4}{x^2+5x-6} \), and the given expression to compare to was also \(\frac{x^2-3x-4}{x^2+5x-6} \). Since the two are equivalent, we can conclude that, yes, the given division is equivalent to \(\frac{x^2-3x-4}{x^2+5x-6} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that allows you to break down equations into simpler parts. It's like finding the building blocks of polynomial expressions. For polynomial division, factoring helps to simplify expressions, making operations like division more straightforward.
To factor a polynomial, you're looking for expressions that multiply together to give you the original polynomial. There are different techniques, like:

• Common factor extraction: Look for a common factor in all terms. For example, in the expression \(x^2 - 3x - 4\), a simple factorization would look for common factors of terms.
• Factoring quadratics: When dealing with quadratics, find two numbers that both multiply to the constant term and add up to the linear coefficient. Like in \(x^2 - 3x - 4\), we need two numbers that multiply to -4 and add up to -3, i.e., \((x - 4)(x + 1)\).

Recognizing patterns in polynomials and applying these techniques helps to simplify them before further operations like division.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Understanding them is crucial because they appear frequently in algebra and calculus.
When working with rational expressions, follow these steps:

• Simplify: Like all fractions, rational expressions can often be simplified by canceling common factors in the numerator and denominator.
• Restrictions on the denominator: Remember that the values that make the denominator zero are not allowed. For example, in a fraction \(\frac{x+1}{x-1}\), \(x\) cannot be equal to 1 because it would make the denominator zero.

The division of two rational expressions also involves finding the reciprocal of the divisor and then multiplying. Just like we rearranged \(\frac{x+1}{x-1} \div \frac{x+6}{x-4}\) to \(\frac{x+1}{x-1} \cdot \frac{x-4}{x+6}\). This method helps in simplifying complex rational expressions efficiently.

Simplification of Fractions

Simplifying fractions is about breaking down expressions to their most basic form by canceling common factors. This makes calculations easier and expressions more manageable.
To simplify fractions:

• Factor numerators and denominators: Break them down to their simplest components, as seen in the polynomial division, like simplifying \(x^2 - 4x + x - 4\) to \((x^2 - 3x - 4)\).
• Cancel common factors: Once factored, cancel any factors that are present in both the numerator and the denominator. This reduces the fraction to a simpler form.

In our example, when multiplying fractions after flipping, \(\frac{x+1}{x-1} \cdot \frac{x-4}{x+6}\) simplifies to \(\frac{x^2-3x-4}{x^2+5x-6}\). This technique is fundamental in ensuring that the expressions you work with are as simple as possible.