Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ 2 m(3 m-4)+m(m-1) $$

Step-by-Step Solution

Answer

The short answer is: \(7m^2 - 9m\).

Step 1: Distribute the coefficients

First, we need to distribute the coefficients to the terms inside the parentheses. For the first term, we'll multiply 2m with both 3m and -4. For the second term, we'll multiply m with both m and -1. \(2m(3m-4)+m(m-1) = 6m^2 - 8m + m^2 - m\)

Step 2: Combine like terms

Next, we'll notice that there are two squared terms (with m^2) and two linear terms (with m). We need to combine these like terms to simplify the expression further. \(6m^2 - 8m + m^2 - m = (6m^2+m^2)+(-8m-m)\) Now, add the coefficients of the like terms: \((6m^2 + m^2) + (-8m - m) = 7m^2 - 9m\) So, the simplified expression is: \(7m^2 - 9m\)

Key Concepts

Distributive Property

Polynomial simplification often begins with the distributive property. This principle allows us to multiply a term outside of the parenthesis by every term inside.
It's useful for breaking down complex expressions. Let's break it down using our example, earlier: \(2m(3m-4)+m(m-1)\).

• First : Multiply \(2m\) with \(3m\) and then \(2m\) with \(-4\).
• This gives you \(6m^2\) and \(-8m\) respectively.
• Second : Multiply \(m\) with \(m\) and then \(m\) with \(-1\).
• This gives \(m^2\) and \(-m\).

When we put all these together, we get \(6m^2 - 8m + m^2 - m\). Following this property ensures that terms are expanded correctly, simplifying the next steps.

Combining Like Terms

The next crucial step in simplifying polynomials is combining like terms. Like terms are terms in an expression that have the same variables raised to the same power. This step shrinks expressions down by merging similar items, making them easier to manage.
In our example, we have like terms: \(6m^2\) and \(m^2\), and another set \(-8m\) and \(-m\).

• For the quadratic terms: Combine \(6m^2\) and \(m^2\) to get \(7m^2\).
• For the linear terms: Combine \(-8m\) and \(-m\) to get \(-9m\).

Now, the combined expression is \(7m^2 - 9m\), which is much simpler. This process is vital to ensure that no errors are made by leaving terms unaddressed.

Algebraic Operations

Algebraic operations such as addition, subtraction, multiplication, and division form the backbone of polynomial simplification. These operations allow us to manipulate polynomial expressions effectively.
The first step is recognizing the type of operation each part of the expression requires. We had addition and multiplication primarily in our solution.

• Multiplication: Used in distributing terms \(2m(3m - 4)\) and \(m(m - 1)\).
• Addition and Subtraction: Once terms were distributed, similar terms were combined, effectively using addition and subtraction to simplify \(6m^2 + m^2\) and \(-8m - m\).

Understanding these operations ensures that we approach each step logically and accurately simplify polynomials.