Problem 80 - Prealgebra and Introductory Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Factorize the algebraic expressions

The first expressions can be factorized as follows: \((x^{2}-3x-4)\) gives \((x-4)(x+1)\) and \((x^{2}+5x-6)\) gives \((x+6)(x-1)\).

Step 2: Rewrite the main fraction

Substituting these factorized expressions into the main fraction, we get: \(\frac{(x-4)(x+1)}{(x+6)(x-1)}\).

Step 3: Simplify the expression

Simplify the expression further by canceling out terms that appear in the numerator and denominator to get \(\frac{x-4}{x+6}\).

Step 4: Compare with the other fraction

The result matches the left-hand side of the division fraction from the original question.

Step 5: Division of fractions

From the knowledge of fractions, the division \(\frac{A}{B} \div \frac{C}{D}\) is equivalent to \(\frac{A}{B} * \frac{D}{C}\). Let's apply this for the provided fractions. Thus \(\frac{x-4}{x+6} \div \frac{x-1}{x+1}\) becomes \(\frac{x-4}{x+6} * \frac{x+1}{x-1}\).

Step 6: Simplify the expression

Distribute the terms that can be multiplied and cancel out like terms. We get \(\frac{x-4}{x+6}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It's all about breaking down a polynomial into simpler components, known as factors, that can be multiplied together to get the original polynomial.
To factor a polynomial, one often looks for ways to express it as a product of its linear or quadratic factors. For example, when we have the polynomial \(x^2 - 3x - 4\), we can rewrite it as \((x - 4)(x + 1)\).
This involves finding two numbers that multiply to give the constant term, \(-4\), and add up to give the coefficient of the linear term, \(-3\).

• Identify the coefficients and constant term.
• Look for factor pairs of the constant term that sum to the linear coefficient.
• Write the polynomial as a product of its factors.

Understanding how to factor polynomials makes it easier to simplify expressions and solve equations. It's like finding the building blocks of a polynomial that can be useful for manipulation or simplification.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing expressions to their most basic form. This is done by eliminating any unnecessary terms and combining like terms to the simplest expression possible.
When simplifying, look for opportunities to factor polynomials, cancel common factors, and combine similar terms.

• Factor polynomials as much as possible.
• Cancel out any common terms in the numerator and denominator.
• Combine like terms to reduce complexity.

In this exercise, once we have the factorized forms, \(\frac{(x-4)(x+1)}{(x+6)(x-1)}\), we can simplify by canceling \((x+1)\) and \((x-1)\) if needed to get the simplest form. Simplification ensures expressions are easier to understand and work with.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Just like regular fractions, they can be added, subtracted, multiplied, and divided.
One must remember that rational expressions need to be simplified as much as possible to make them manageable.

• Always factor the polynomials in the numerator and the denominator.
• Simplify by canceling out common factors between them.
• Be cautious of restrictions, such as values that make the denominator zero.

In the provided exercise, we start with \(\frac{x^2-3x-4}{x^2+5x-6}\), and find that it simplifies to the rational expression \(\frac{x-4}{x+6}\) after proper factoring and simplification. This new expression is equivalent to the result obtained after performing the division operation. Working with rational expressions involves similar techniques as handling regular fractions, with added attention to the properties of polynomials.