Problem 40 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{x^{2}+x-12}{x^{2}-4 x+3} $$

Step-by-Step Solution

Answer

Question: Reduce the following rational expression to its lowest terms: \(\frac{x^2 + x - 12}{x^2 - 4x + 3}\) Answer: \(\frac{x+4}{x-1}\)

Step 1: Factor the numerator and denominator

First, we need to factor both the numerator and the denominator of the given rational expression. The numerator, \(x^2 + x - 12\), can be factored as \((x+4)(x-3)\). The denominator, \(x^2 - 4x + 3\), can be factored as \((x-3)(x-1)\). So, the original rational expression can be written as: $$ \frac{(x+4)(x-3)}{(x-3)(x-1)} $$

Step 2: Cancel the common factors

Now we need to cancel the common factors in the numerator and the denominator. The common factor between the numerator and the denominator is \((x-3)\). So, we can cancel this factor in both the numerator and the denominator. After canceling the common factor, we get: $$ \frac{(x+4)}{(x-1)} $$

Step 3: Write the simplified expression

Having canceled the common factors, we now have the simplified rational expression: $$ \frac{x+4}{x-1} $$ The given rational expression has been reduced to its lowest terms.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. When you are faced with expressions like \(x^2 + x - 12\), your goal is to write it as a product of simpler expressions. This process involves finding two binomials that multiply to give you the original quadratic expression. To factor any quadratic expression of the form \(ax^2 + bx + c\), you need to find two numbers that add to \(b\) and multiply to \(ac\). In the case of \(x^2 + x - 12\), the numbers \(4\) and \(-3\) work because they add up to \(1\) (the coefficient of \(x\)) and multiply to \(-12\) (the constant term). Thus, the expression factors to \((x+4)(x-3)\).
By applying similar principles, we factor the denominator \(x^2 - 4x + 3\) into \((x-3)(x-1)\). Factoring makes it easier to simplify expressions further, so it's an essential first step in many algebra problems.

Simplifying Fractions

Simplifying fractions involves expressing a fraction in its simplest form, where the numerator and the denominator have no common factors other than 1. For rational expressions, this means identifying and canceling common factors between the numerator and the denominator.
After factoring both the numerator and the denominator, look for terms that appear in both to simplify them. For instance, in the rational expression \(\frac{(x+4)(x-3)}{(x-3)(x-1)}\), the term \((x-3)\) appears in both the numerator and the denominator. Once these are identified, they can be canceled out, leaving you with \(\frac{x+4}{x-1}\). This process of simplifying fractions by canceling common factors makes expressions much easier to work with.

Reducing Expressions to Lowest Terms

Reducing a rational expression to its lowest terms means simplifying it as much as possible. It involves several steps: factoring, simplifying by canceling common factors, and rewriting the expression without the canceled parts. The key is to ensure that there are no common factors remaining between the numerator and the denominator.
Returning to the example given, we started with \(\frac{x^2+x-12}{x^2-4x+3}\). After factoring and canceling the common factor \((x-3)\), we simplified it to \(\frac{x+4}{x-1}\). This simplified expression is now in its lowest terms because \(x+4\) and \(x-1\) do not share any common factors. Reducing expressions to lowest terms not only simplifies the problem but also helps avoid errors in calculations down the line.