Problem 24 - Algebra 1: Concepts and Skills

Question

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{x}{x^{2}+5 x-24}+\frac{8}{x^{2}+5 x-24} $$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{x}{x^{2}+5x-24}+\frac{8}{x^{2}+5x-24}\) is \(\frac{x+8}{x^{2}+5x-24}\).

Step 1: Review the Given Expressions

We have two fractions: \(\frac{x}{x^{2}+5x-24}\) and \(\frac{8}{x^{2}+5x-24}\). Both fractions have the same denominator, so they are ready to be combined.

Step 2: Combine the Fractions

Since the two fractions have the same denominator, we can add them together by adding their numerators: \(\frac{x+8}{x^{2}+5x-24} = \frac{x+8}{x^{2}+5x-24}\). Thus, we have combined the two fractions into a single fraction.

Step 3: Simplify the Expression

At this point, the fraction we have resulted, \(\frac{x+8}{x^{2}+5x-24}\), is already in its simplest form. There are no common factors between the numerator and the denominator, so no further simplification is possible.

Key Concepts

Simplifying Algebraic Fractions

In algebra, the process of making a complex expression into a simpler, more manageable form is essential, and this is where simplifying algebraic fractions comes into play. An algebraic fraction is similar to a numeric fraction, but it contains variables like 'x' or 'y'. Simplification can involve reducing the fraction to its lowest terms by canceling out common factors in the numerator and denominator.

To simplify an algebraic fraction, we look for common factors that can be divided out. In the textbook problem, the expression of interest is \[\frac{x+8}{x^{2}+5x-24}\]. In this scenario, simplifying might entail factoring the quadratic expression in the denominator to see if any factors cancel out with the numerator. However, since there are no common factors between the numerator and the factorized form of the denominator (which would be \(x+8\) and \(x-3\) if factored), the expression is in its simplest form.

Combining Like Terms

Algebra often presents scenarios where you need to combine like terms to simplify expressions. Like terms are terms that have the same variable raised to the same power. In the example problem, when adding \(\frac{x}{x^{2}+5x-24}\) and \(\frac{8}{x^{2}+5x-24}\), they are considered like terms, because they both share the same denominator. Combining them involves adding the numerators while keeping the denominator constant.

This is different from the initial step of simplifying, as combining like terms doesn't necessarily make the fraction simpler, but it does transform it into a single term that is easier to work with. After combining, you check again for any simplifications, looking for common factors that can now be possibly canceled out.

Algebraic Fraction Addition and Subtraction

Understanding algebraic fraction addition and subtraction is crucial when multiple fractions with algebraic expressions are involved. The key factor that determines ease of addition or subtraction is having a common denominator. If the fractions do not have a common denominator, you must first obtain one by finding the least common multiple of the denominators.

In our exercise, the two fractions \(\frac{x}{x^{2}+5x-24}\) and \(\frac{8}{x^{2}+5x-24}\) are already equipped with a common denominator, which simplifies the process. The numerators are added directly to find the sum. This is a straightforward process in this case, but with differing denominators, one would have to manipulate the fractions to achieve a common denominator before combining. It's a good practice to always look for simplification opportunities after completing the addition or subtraction.