Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Write the rational expression in simplest form.\(\frac{x+1}{x^{2}-3 x-4}\)

Step-by-Step Solution

Answer

The simpliest form of the given expression is \( \frac {1}{x-4}\).

Step 1: Factorize the denominator

The denominator is a quadratic equation of the form \(ax^2+bx+c\), where \(a=1\), \(b=-3\), and \(c=-4\). To factorize this, search for two numbers that add up to \(b=-3\), and multiply to \(c=-4\). The numbers are -4 and 1, because -4 + 1 = -3 and -4 * 1 = -4. So, the factorized form of \(x^2 - 3x - 4\) is \((x - 4)(x + 1)\). Hence, the expression becomes \( \frac {x+1}{(x-4)(x+1)}\).

Step 2: Simplifying the expression

Looking at the numerator and denominator of the expression, there's a common term \((x+1)\) which can be cancelled out. By doing that, the expression becomes \( \frac {1}{x-4}\).

Key Concepts

Factoring Quadratics

The process of factoring quadratics is essential when working with rational expressions because it simplifies complex expressions to a more manageable form. Imagine you have a quadratic equation like \(ax^2 + bx + c\). This equation is in polynomial form and needs to be broken down into simpler products of binomials. In simpler terms, you are trying to express it as the product of two simpler expressions.

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic expression. In the expression \(x^2 - 3x - 4\), \(a = 1\), \(b = -3\), and \(c = -4\).
• Look for two numbers that add to \(b\) and multiply to \(c\). Here, -4 and 1 fit because -4 + 1 = -3 and -4 * 1 = -4.

Break the quadratic into two binomials using these numbers: \(x^2 - 3x - 4 = (x - 4)(x + 1)\). Factoring quadratics is like finding puzzle pieces that fit together to form a coherent picture. When you correctly factor these expressions, it allows you to see common elements that simplify more complex algebraic expressions.

Simplifying Algebraic Expressions

Algebraic expression simplification is about making expressions as straightforward as possible. This can be done once we've factored any part of the expression that can be simplified. For rational expressions, post-factoring, check to see if any terms appear in both the numerator and the denominator.For instance, after factoring, the expression \(\frac{x+1}{(x-4)(x+1)}\) has \(x+1\) in both parts. They can be simplified, or in other words, cancelled out. Removing these duplicate terms helps boil down the expression to its simplest form.

• Identify common factors between the numerator and the denominator.
• Simplify by canceling these common factors.

In our example, after performing this step, the simplified expression becomes \(\frac{1}{x-4}\). Simplifying makes math more straightforward and helps in finding solutions quickly without being bogged down by larger expressions.

Canceling Common Factors

Canceling common factors is a key strategy in simplifying rational expressions. This involves identifying and removing factors that appear in both the numerator and the denominator. By cancelling these factors, you reduce the expression to its simplest form.To visualize, consider you have two expressions written as a fraction. If the same element \(x + 1\) appears in both the numerator and the denominator of \(\frac{x+1}{(x-4)(x+1)}\), it can be removed through cancellation. This step is only valid when the factor appears in both parts of the fraction.

• Cross out identical terms from the numerator and denominator.
• Ensure that what remains is simpler and more concise.

This step acts like pruning a tree, removing the excess leaves, revealing a clear and neat structure. What's left after canceling gives the rational expression \(\frac{1}{x-4}\), significantly simpler, making it much easier to work with or solve.