Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ \left(2.4 x^{3}-3 x^{2}+1.7 x-6.2\right)-\left(1.2 x^{3}+1.2 x^{2}-0.8 x+2\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(1.2 x^{3} - 4.2 x^{2} + 2.5 x - 8.2\).

Step 1: Identify the corresponding terms

The corresponding terms are the terms that have the same variable and exponent in both expressions. In this exercise, the corresponding terms are: - \(2.4 x^{3}\) and \(1.2 x^{3}\) - \(-3 x^{2}\) and \(1.2 x^{2}\) - \(1.7 x\) and \(-0.8 x\) - \(-6.2\) and \(2\)

Step 2: Subtract the corresponding terms

Now, subtract the corresponding terms: - \(2.4 x^{3} - 1.2 x^{3} = (2.4 - 1.2) x^{3} = 1.2 x^{3}\) - \(-3 x^{2} - 1.2 x^{2} = (-3 - 1.2) x^{2} = -4.2 x^{2}\) - \(1.7 x - (-0.8 x) = (1.7 + 0.8) x = 2.5 x\) - \(-6.2 - 2 = -6.2 - 2 = -8.2\)

Step 3: Combine the results

Finally, combine the simplified terms to get the final answer: \(1.2 x^{3} - 4.2 x^{2} + 2.5 x - 8.2\) So, the simplified expression is \(1.2 x^{3} - 4.2 x^{2} + 2.5 x - 8.2\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a foundational skill in algebra. It involves reducing an expression to its simplest form, making it easier to handle and understand. When you're faced with a clutter of numbers and variables, simplifying can seem daunting, but by following systematic steps, the process becomes straightforward.

Take the given polynomial expression, for example. The first step is to identify like terms which are grouped by shared variables and exponents. Once identified, you can perform arithmetic operations—adding or subtracting coefficients—to simplify the expression. In our case, when subtracting polynomials, remember to change the sign of the second polynomial's terms before the combination, as subtraction can be thought of as adding a negative.

Combining Like Terms

Combining like terms is the bread and butter of algebraic simplification. Like terms are expressions that contain the same variable raised to the same power. They're essentially part of the same family and can be added or subtracted from one another.

In our exercise, you'll notice terms with the same variable and exponent, such as \(2.4 x^{3}\) and \(1.2 x^{3}\), which can be combined through subtraction since we're dealing with polynomial subtraction here. This process shrinks the expression down to a more manageable size, keeping it neat and tidy. Consistently applying the appropriate arithmetic to each pair of like terms brings us closer to a single polished expression.

Algebraic Operations

Engaging with algebraic operations involves becoming fluent with the basic building blocks of algebra: addition, subtraction, multiplication, and division. When dealing with polynomial subtraction, as in the given exercise, we follow specific rules. The subtraction operation between polynomials is performed term by term. This means that we align each term from one polynomial with its corresponding counterpart in the other, and subtract them individually.

Another key point is remembering to subtract the entire term, not just the coefficient. This might involve changing the signs when distributing the subtraction across terms, as is effectively illustrated in the step-by-step solution provided. The ordered and logical approach to these operations is crucial for accurate algebraic manipulation.

Applied Mathematics

Applied mathematics is the use of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In other words, it's when the rubber meets the road in math. The principles of algebra, especially polynomial operations, are fundamental in various applied mathematics scenarios.

For example, polynomial expressions can represent trajectories in physics, investment growth over time in finance, or even the intensity of light across a surface in computer graphics. Understanding how to manipulate these expressions through addition, subtraction, and other operations therefore has real-world implications. Thus, exercises like these, far from being merely academic, arm students with the ability to solve practical problems outside the classroom.