Problem 54 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-7 x+6} $$

Step-by-Step Solution

Answer

The simplified expression is \(- \frac{5x}{(x-6)(x+4)(x-1)}\).

Step 1: Simplify the Denominators

The denominators of both fractions can be factored as follows: \[x^{2}-2 x-24 = (x-6)(x+4)\]and\[x^{2}-7 x+6 = (x-1)(x-6)\]So the fractions become:\[ \frac{x}{(x-6)(x+4)} - \frac{x}{(x-1)(x-6)}\]

Step 2: Find the Common Denominator

In order to add or subtract fractions, you first need a common denominator. Here, the common denominator would be the product of all distinct factors in the denominator of both fractions. Thus, the common denominator is \((x-6)(x+4)(x-1)\).

Step 3: Perform Subtraction

Using the common denominator, rewrite each fraction with the common denominator and perform the subtraction. The expression becomes:\[ \frac{x(x-1) - x(x+4)}{(x-6)(x+4)(x-1)}\]which simplifies further to:\[ \frac{x^{2}-x - x^{2}-4x}{(x-6)(x+4)(x-1)}\]and finally:\[ \frac{-5x}{(x-6)(x+4)(x-1)}\]

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is essential when working with algebraic fractions. It allows us to break down complex polynomial expressions into simpler, multiplied forms. Factoring often makes it easier to see common elements and simplify expressions further, just like we did with the denominators of the given fractions.

To factor a polynomial, you’re looking to express it as a product of its factors. For example, take the polynomial \(x^2 - 2x - 24\). This polynomial can be rewritten as \((x-6)(x+4)\) by finding two numbers that multiply to

• The constant term (-24) and
• Add up to the linear coefficient (-2).

This same principle was used to factor \(x^2 - 7x + 6\), resulting in \((x-1)(x-6)\). Once factored, we gained a clearer picture of the problem and transformed it into one that is easier to manage.

Common Denominator

The concept of a common denominator is crucial when adding or subtracting fractions. It refers to a denominator that is shared by all fractions involved. Having a common denominator allows you to combine the fractions without changing their values.

In the exercise, the original fractions' denominators were factored, and the common denominator was determined as \((x-6)(x+4)(x-1)\). This involves taking all distinct factors from each denominator and multiplying them together. The common denominator must include each factor as many times as it appears in any of the denominators.

This meticulous process ensures that the two algebraic fractions can be rewritten with the same denominator, paving the way to perform addition or subtraction properly.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. Once common denominators are established, you can simplify the expression by performing algebraic operations. This often includes combining like terms and canceling common factors.

In this exercise, after rewriting each fraction using the common denominator, subtraction was performed:

• Each numerator expression was expanded: \(x(x-1) - x(x+4)\).
• Simplification followed: \(x^2 - x - x^2 - 4x = -5x\).

The resulting fraction, \(\frac{-5x}{(x-6)(x+4)(x-1)}\), is in a simplified form where the expression is as reduced as possible without losing equivalency. Simplifying helps in understanding and interpreting expressions more efficiently, which is beneficial for solving and analyzing mathematical problems.