Problem 61 - Algebra A Combined Function

Question

Then list four equivalent forms for each rational expression. $$ -\frac{x-10}{x+8} $$

Step-by-Step Solution

Answer

Equivalent forms: \(\frac{x-10}{x+8}\), \(\frac{10-x}{8+x}\), \(\frac{10-x}{x+8}\), and \(-\frac{x-10}{x+8}\).

Step 1: Identifying the Expression

The given expression is a rational expression: \[-\frac{x-10}{x+8}\].This expression consists of a numerator \(x-10\) and a denominator \(x+8\), with a negative sign in front of the fraction.

Step 2: Equivalent Form - Multiply Numerator by -1

To generate an equivalent expression, we can multiply the numerator by -1, and simultaneously multiply the negative sign outside by -1, converting it into a positive sign:\[\frac{-(x-10) imes -1}{x+8} = \frac{x-10}{x+8}\].Hence, an equivalent form is \[\frac{x-10}{x+8}\].

Step 3: Equivalent Form - Multiply Denominator by -1

Next, multiply both the numerator and the denominator by -1, effectively switching the signs inside:\[-\frac{-(x-10)}{-(x+8)} = \frac{-x+10}{-x-8} = \frac{10-x}{-x-8}\].Thus, another equivalent form is:\[\frac{10-x}{8+x}\].

Step 4: Equivalent Form - Factor out -1 from Numerator

Factor out a -1 from the numerator:\[-\frac{x-10}{x+8} = \frac{-(x-10)}{x+8} = \frac{10-x}{x+8}\].This provides us with a third equivalent form:\[\frac{10-x}{x+8}\].

Step 5: Use a Common Polynomial Factor

Introduce a common polynomial factor such as \(1\) as \(x-x\), or \((x-5) + 5\), simplifying through distribution. As the form obtained from Step 4 was:\[\frac{10-x}{x+8}\], this gives us our fourth equivalent form:\[\frac{10+(0-x)}{8+(x-x)+x}\], simplifying as \[-\frac{x-10}{x+8}\], or just rearranging.Hence equivalent form:\[-(x-10)/(x+8)\].

Key Concepts

Equivalent Forms

Rational expressions can be transformed into different but equivalent forms through various algebraic manipulations. The key is to alter the expression without changing its overall value. In this exercise, we learned how various operations like multiplication by -1, distribution, or factoring can result in expressions that look different but are mathematically equivalent to the original.

Some methods include:

• Multiplying the entire expression by -1.
• Switching signs in the numerator or denominator.
• Rearranging terms or simplifying complex parts using polynomial identities.

Understanding these transformations is crucial as it allows one to simplify complex expressions, making it easier to handle or solve problems involving rational expressions.

Numerators and Denominators

In any rational expression, such as \(-\frac{x-10}{x+8}\), the numerator and denominator play distinct roles. The numerator is the expression above the division line, and the denominator lies below.

Some essential points include:

• Operations such as multiplication can be done separately on numerators and denominators as long as they're done consistently on both.
• Signs (positive or negative) in front of numerators or denominators affect the entire expression's value.
• Manipulating numerators and denominators is central to solving or simplifying rational expressions.

Rational expressions inherently depend on the relationship between the numerator and the denominator. Therefore, understanding how they interact and can be manipulated is crucial in working with these mathematical forms.

Factoring

Factoring is an invaluable skill in algebra, especially when dealing with polynomial expressions in rational forms. It involves breaking down a polynomial into simpler terms (or factors) that multiply together to create the original expression.

In this exercise, factoring was used by:

• Identifying and applying common factors in numerators or denominators.
• Factoring out a \(-1\) to reverse signs and simplify expressions.

By finding factors, we simplify complex expressions and transform them into equivalent forms, facilitating easier computation or comparison. Recognizing and utilizing these factors allows us to manipulate expressions effectively.

Polynomial Expressions

Polynomial expressions are algebraic expressions involving variables raised to whole-number exponents, potentially including constant terms. Rational expressions often contain polynomials as their numerators or denominators.

Key characteristics include:

• They consist of terms that are added or subtracted.
• Polynomial expressions can often be rewritten through factoring or other algebraic techniques.
• Simplification involves distributing, combining like terms, and reworking polynomial components.

In this activity, polynomial manipulation was at the core of identifying equivalent rational expressions. Understanding polynomial expressions is fundamental in making algebraic manipulations, solving equations, and simplifying rational expressions possible.