Problem 16 - A Graphical Approach to College Algebra

Question

Write each rational expression in lowest terms. $$\frac{36 y^{2}+72 y}{9 y}$$

Step-by-Step Solution

Answer

\(4(y+2)\)

Step 1: Factor the Numerator

The numerator is given by the expression \(36y^2 + 72y\). Notice that both terms have a common factor. Factor out the greatest common factor (GCF), which is \(36y\), from the numerator: \[36y^2 + 72y = 36y(y + 2)\].

Step 2: Simplify the Expression

Write the rational expression using the factored numerator:\[\frac{36y(y+2)}{9y}\].Next, divide both the numerator and the denominator by the greatest common factor they share, which is \(9y\): \[\frac{36y(y+2)}{9y} = \frac{36(y+2)}{9}\].

Step 3: Divide to Simplify Further

Now, focus on the numerical coefficients. Divide 36 by 9:\[\frac{36(y+2)}{9} = \frac{4(y+2)}{1} = 4(y+2)\].

Step 4: Write the Final Result

The expression is now completely simplified. It is in its lowest terms:\[4(y+2)\].

Key Concepts

Simplifying Expressions

Simplifying rational expressions means rewriting an expression in its simplest form. This process helps make complex expressions more manageable and easier to work with. Simplifying is all about reducing the fraction to its most basic form, much like how you would simplify a numerical fraction.

In the given exercise, we started with the expression \( \frac{36y^2 + 72y}{9y} \). We first factor the numerator and the denominator to find the greatest common factor between them. After identifying the common factors, we cancel them out to simplify the expression. This is similar to reducing \(\frac{18}{27}\) by dividing both the numerator and the denominator by their greatest common factor, 9, resulting in \(\frac{2}{3}\).

The objective with rational expressions is the same: reduce the expression as much as possible by finding and canceling out common factors. This makes them easier to analyze and solve.

Common Factor

A common factor is a number or an expression that divides two or more numbers or expressions exactly, without leaving a remainder. Identifying common factors is the key step in simplifying any expression, especially when dealing with algebraic terms.

In our exercise, the terms in the numerator \(36y^2 + 72y\) share a common factor. To find this factor, we look for the greatest common factor (GCF) of both terms—a concept which involves factoring both terms down to their prime components. For instance, the GCF of 36 and 72 is 36, as 36 divides both numbers without any remainder. Similarly, both terms share the variable \(y\).

Thus, the GCF of the expression is \(36y\), allowing us to rewrite the numerator as \(36y(y + 2)\). This process of finding a common factor is crucial in simplifying rational expressions, enabling easy cancellation that leads to a reduced form.

Factoring

Factoring is a process where we express a mathematical expression as a product of its factors. When working with algebraic expressions, factoring becomes essential for simplification, solving equations, and even for graphing functions.

In the original expression \(36y^2 + 72y\), factoring is used to take out common elements, making the expression simpler. Using the greatest common factor, 36y, we rewrite the expression as \(36y(y + 2)\). This step is essential because it allows us to then cancel terms from both the numerator and the denominator. Factoring helps us break down complex pieces into simpler, more solvable parts.

Factoring is like finding building blocks; instead of handling bulky blocks of bricks, they are reduced to more manageable sizes. In algebra, these building blocks (or factors) enable us to solve larger problems by focusing on smaller, simpler components.