Problem 104 - Introductory Algebra for College Students

Question

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\text { Simplify: } \frac{(y+2) y-2 \cdot 4}{4 y(y+4)}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y-2}{4y}\)

Step 1: Simplify the numerator

First, distribute the \(y\) inside the parentheses \(y*(y + 2)\) and then subtract \(2*4\) from it. This results to \( y^2 + 2y - 8 \)

Step 2: Simplify the denominator

Multiply \(4y\) and \(y+4\) to get \(4y^2 + 16y\) in the denominator.

Step 3: Divide the simplified numerator by the simplified denominator

Now, divide the simplified numerator by the simplified denominator, \(\frac{y^2 + 2y - 8}{4y^2 + 16y}\)

Step 4: Factor the numerator and denominator

Factor the numerator to \((y-2)*(y+4)\) and denominator to \(4y*(y + 4)\)

Step 5: Simplify the expression

Now, cancel common factors in the numerator and denominator. The \(y+4\) term in the numerator and denominator cancel out each other, leaving the simplified expression as \(\frac{y-2}{4y}\)

Key Concepts

Factoring

Factoring is a crucial skill when working with rational expressions because it helps simplify complex equations. When you encounter a polynomial, such as in this exercise, your job is to express it as a product of simpler polynomials or numbers. This means finding the factors of expressions like the numerator and denominator in rational expressions.

In the given exercise, the numerator was initially expressed as \(y^2 + 2y - 8\). The goal was to express it in its factored form, \((y - 2)(y + 4)\). Factoring plays a similar role for the denominator, initially \(4y^2 + 16y\), which eventually became \(4y(y + 4)\).

Factoring often involves common methods such as:

• Looking for common factors
• Using the difference of squares
• Utilizing the quadratic formula for more complex expressions

Being proficient in factoring will aid in making further steps, like simplification, much easier.

Numerator and Denominator

In any rational expression, the structure is a fraction where the numerator is the expression on the top, and the denominator is the one at the bottom. Both play distinct roles in the simplification process.

The numerator dictates the overall value of the fraction. In the exercise, the initial numerator, \((y+2)y - 2 \cdot 4\), contained various terms needing simplification before factoring. Similarly, the denominator \(4y(y+4)\) determines the fraction's divisor.

Caring for both parts is essential:

• First, simplify each part individually by carrying out operations such as distribution or multiplication.
• Next, factor out expressions within the numerator and denominator to enable simplification.

This allows easier management and adjustment of the fraction as a whole.

Simplification

Simplification is the process of reducing a fraction or expression to its simplest form. This includes breaking down complex terms into more manageable ones to make the expression clearer and often involves working with both the numerator and denominator together.

In our exercise, simplification occurred at multiple levels:

• First, by distributing the products within the numerator and denominator to form expressions like \(y^2 + 2y - 8\) and \(4y^2 + 16y\).
• Then, by factoring these expressions, simplifying them further led to \((y-2)(y+4)\) and \(4y(y+4)\) in the denominator.

The end goal of simplification is to facilitate easier calculations and interpretations, which is achieved once both parts are presented in their most reduced form.

Canceling Common Factors

Canceling common factors is a technique used to simplify rational expressions by removing identical factors from both the numerator and the denominator.

When both parts of the fraction share a factor, it can be "canceled," which means it can be divided out of both parts, reducing the expression. In the exercise, after factoring the numerator and denominator:

• The common factor \((y+4)\) was identified in both the numerator and the denominator.
• This allowed \((y+4)\) to be canceled from the fraction, leaving the simpler expression \(\frac{y-2}{4y}\).

Canceling common factors is only valid when the factor being canceled is not zero since division by zero is undefined. This step is vital for ensuring equations are reduced to their lowest terms, making further calculations straightforward and concise.