Question

Write the rational expression in simplest form.\(\frac{x-5}{10-2 x}\)

Short answer

The simplest form of the given rational expression is \(\frac{5-x}{2x-10}\)

Step-by-step solution

Rewrite the denominator in standard form

To simplify the expression, we first rewrite the denominator in standard form: \(10-2x = -2x + 10\)

Factor out the common factor that exists in the denominator

We factor the denominator as -1(2x-10). So, the expression becomes \(\frac{x-5}{-1(2x -10)}\)

Simplify the denominator

Divide every term of the expression by -1. Then the expression becomes \(-\frac{x-5}{2x -10}\)

Simplify further

We divide the numerator and the denominator each by -1 to clear the negative sign in the denominator. Then the expression becomes\(-(-\frac{x-5}{2x-10}) = \frac{5-x}{2x-10}\)

Key concepts

Simplification of Expressions

Simplifying expressions is about reducing them to their most basic form. It makes them easier to understand and work with. Simplification can include combining like terms, reducing fractions, or factoring polynomial expressions. In rational expressions, we specifically aim to simplify the ratio of two polynomials.

For instance, when given the expression \( \frac{x-5}{10-2x} \), simplification allows us to rewrite it in a simpler form by manipulating its components. A critical part of simplification is identifying opportunities for cancellation, where common terms in the numerator and the denominator can be divided out. This doesn't change the expression's value but changes its appearance to a simpler equivalent. Remember, simplifying isn't just about making things smaller; it's about clarity and ease of use.

Factoring

Factoring is the process of breaking down a mathematical expression into its simplest pieces, known as factors, which when multiplied, result in the original expression. In the context of rational expressions, factoring plays a crucial role in simplification.

Take the expression, \(10-2x\), for example. We can factor it by looking for the greatest common factor, which, in this case, is \(-2\). Factoring out \(-2\) gives us \(-2(x-5)\). This step reveals the underlying structure of the expression, allowing for potential simplification because the term \(x-5\) appears in both the numerator and the denominator. Remember that successful simplification often hinges on effective factoring.

Negative Signs Handling

Negative signs in algebra can change the entire meaning of an expression, so handling them correctly is key. They can often be a source of confusion, but with careful attention, you can manage them effectively.

In the expression \(\frac{x-5}{-2(x-5)}\), the negative sign can be factored out of the entire denominator, turning the expression into \(-\frac{x-5}{2(x-5)}\). By distributing or factoring out the negative correctly, we maintain the mathematical equivalence of the expression while simplifying its form. If both the numerator and denominator are negative, dividing them both by \(-1\) simplifies things further, which is what enabled us to reach \(\frac{5-x}{2x-10}\) ultimately.

This step underscores the importance of negative signs when simplifying, as they can be manipulated to make expressions more streamlined without affecting their value.

College Algebra

College algebra encompasses a wide array of topics aimed at building a solid mathematical foundation. It prepares students for advanced studies in mathematics and related fields.

Simplifying rational expressions is a common topic in college algebra. It involves many fundamental techniques such as factoring, distributing, and handling negative signs, all of which enhance problem-solving skills.

Mastering these skills in college algebra is crucial because they are not just academic exercises but tools that are frequently used in various mathematical applications, from calculus to real-world problem-solving scenarios. Simplifying rational expressions contextually illustrates the interconnectedness of different algebraic concepts and is an excellent way to deepen one's understanding of mathematics.