Problem 64 - Algebra A Combined Function

Question

Then list four equivalent forms for each rational expression. $$ -\frac{8 y-1}{y-15} $$

Step-by-Step Solution

Answer

Equivalent forms: \(\frac{-(8y-1)}{y-15}\), \(\frac{-16y+2}{2y-30}\), \(\frac{-24y+3}{3y-45}\), \(\frac{-8y^2+y}{y^2-15y}\).

Step 1: Understand the rational expression

The given rational expression is \(-\frac{8y-1}{y-15}\). A rational expression is simply a fraction where the numerator and the denominator are polynomials.

Step 2: Multiply by -1 to Change Signs

Since the given expression is \(-\frac{8y-1}{y-15}\), we can remove the negative sign by rewriting it as \(\frac{-(8y-1)}{y-15}\). This is an equivalent form.

Step 3: Multiply Numerator and Denominator by the Same Non-zero Number

We can multiply both the numerator and the denominator by any non-zero real number to get another equivalent form. For example, multiplying by 2 gives us \(\frac{2(-8y+1)}{2(y-15)} = \frac{-16y+2}{2y-30}\).

Step 4: Factor out Constants

Factor out constants from the numerator. The original numerator \(-8y+1\) does not allow for additional factoring, so let's multiply it by 3 for this example: \(\frac{-3(8y-1)}{3(y-15)} = \frac{-24y+3}{3y-45}\).

Step 5: Multiply by a Variable Term

Multiply both the numerator and denominator by y, a common variable that doesn't change the value of a fraction. This gives us \(\frac{y(-8y+1)}{y(y-15)} = \frac{-8y^2+y}{y^2-15y}\).

Key Concepts

Equivalent Forms of Expressions

Expressions in mathematics can often be expressed in multiple ways while maintaining the same value. For rational expressions, which are essentially fractions involving polynomials, finding equivalent forms often involves manipulating both the numerator and the denominator without changing the expression's value.

To create equivalent forms, you can perform operations that affect both the numerator and the denominator in the same manner. Here are a few approaches:

- Changing the sign: By multiplying the entire rational expression by -1, you change the signs of the terms but not the value. - Scaling: Multiplying both the numerator and the denominator by the same non-zero number will result in a new but equivalent expression. - Introducing variables: Multiplying by a variable term that simplifies appropriately when expanded helps to form equivalent expressions.

By employing these techniques, you can transform an expression into different forms that may be easier to work with in specific contexts.

Factoring in Algebra

Factoring is a key concept in algebra and involves breaking down an expression into a product of simpler expressions. When dealing with polynomial fractions, factoring can help simplify the expression further.

Algebraic factoring often involves:

- Common factors: Identify and divide out the greatest common factor from the terms in the expression. - Factoring trinomials: Rewrite trinomials as a product of binomials. - Difference of squares: Recognize expressions of the form \( a^2 - b^2 \) and rewrite as \((a-b)(a+b)\).

While the given rational expression \(-\frac{8y-1}{y-15}\) does not allow for straightforward factoring of its parts as they are simple linear terms, understanding these factoring methods is crucial for other more complex polynomial fractions. Factoring is an essential step in simplifying expressions and solving polynomial equations efficiently.

Polynomial Fractions

Polynomial fractions are those where both the numerator and the denominator are polynomial expressions. They represent a subset of rational expressions. Understanding how to manipulate these fractions is important for simplification and solving algebra problems.

Key points to consider include:

- Simplification: Checking if the numerator and the denominator share factors that can be canceled out. - Operations: Adding, subtracting, multiplying, or dividing polynomial fractions by finding a common denominator or factoring. - Handling variables: Dealing with the variable terms properly, ensuring that equivalent expressions are correctly formed by multiplying or dividing by appropriate factors.

Learning to work with polynomial fractions makes handling complex mathematical tasks more manageable while paving the way for understanding deeper algebraic concepts and functions.