Problem 57 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The strength of a human body's reaction to a dosage \(D\) of a certain drug, where \(k\) is a positive constant, is given by $$\frac{k D^{2}}{2}-\frac{D^{3}}{3}$$ Factor this expression.

Step-by-Step Solution

Answer

The factored expression is given by \(D^2 \left(\frac{3k - 2D}{6}\right)\).

Step 1: Factor out the GCF

In this case, the greatest common factor (GCF) is a factor of both terms of the expression. Observing the expression, we notice that both terms have the factor \(D^2\). We will now factor out \(D^2\): \[D^2 \left(\frac{k}{2} - \frac{D}{3}\right)\]

Step 2: Simplify the expression inside the parentheses

Now, let's simplify the expression inside the parentheses. In this case, we can write both terms with a common denominator, which is 6: \[D^2 \left(\frac{3k}{6} - \frac{2D}{6}\right)\] Finally, our factored expression is: \[D^2 \left(\frac{3k - 2D}{6}\right)\]

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables represent unknown values and are often denoted by letters, such as x, y, or k. For example, the expression \( \frac{k D^{2}}{2}-\frac{D^{3}}{3} \) from the exercise is an algebraic expression involving the variable D which represents the dosage, and the constant k.

When working with algebraic expressions, it's crucial to understand how to manipulate and simplify them. This involves combining like terms, factoring, expanding, and sometimes, performing more complex operations like completing the square or applying the quadratic formula. In our case, we are interested in factoring, which is a method of simplifying expressions to make them easier to work with or to solve equations.

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the biggest factor that divides two or more numbers or terms. To spot the GCF, look for the highest power of the common variables and constants that are present in every term of the algebraic expression.

For instance, in our exercise, both terms \( \frac{k D^{2}}{2} \) and \( \frac{D^{3}}{3} \) have the variable D in common. The GCF here is \( D^2 \) because it's the highest power of D that is present in both terms. Factoring out the GCF is often the first step in simplifying an algebraic expression and is essential in understanding how to break down and reassemble complex expressions.

Simplifying Expressions

The process of simplifying expressions reduces them to their simplest form, making calculations and understanding the relationship between variables more manageable. One way to simplify an expression is to factor out the GCF, like we did in the exercise.

Once the GCF is factored out, the remaining expression is easier to deal with. In our example, factoring out \( D^2 \) led us to work within the parentheses, \( \frac{k}{2} - \frac{D}{3} \) which we then simplified further by finding a common denominator. This step ensures the expression inside the parentheses is in its simplest form, resulting in the final simplified expression \( D^2 \left(\frac{3k - 2D}{6}\right) \) which is easier to evaluate or use in subsequent algebraic operations.

Simplifying expressions is not just about making them shorter; it can also reveal certain properties of the algebraic expression that may not have been apparent in its original form. It is a fundamental skill in algebra that helps with solving equations and inequalities.