Problem 36 - College Algebra with Modeling and Visualization

Question

Subtract the polynomials. $$(10 x+5)-(-6 x-4)$$

Step-by-Step Solution

Answer

The result is \(16x + 9\).

Step 1: Identify the Polynomials

We are given the expression \((10x + 5) - (-6x - 4)\). Here, \(P_1 = 10x + 5\) and \(P_2 = -6x - 4\).

Step 2: Distribute the Negative Sign

Subtracting \(-P_2\) is equivalent to adding \(P_2\). Therefore, this expression becomes \(10x + 5 + 6x + 4\).

Step 3: Combine Like Terms

Identify and combine the like terms in the expression. The like terms \(10x\) and \(6x\) combine to \((10 + 6)x = 16x\). The constants \(5\) and \(4\) combine to \(5 + 4 = 9\).

Step 4: Write the Result

After combining like terms, the simplified expression is \(16x + 9\).

Key Concepts

Combining Like Terms

Combining like terms is a crucial concept when working with algebraic expressions. It refers to the process of simplifying an expression by merging terms that have the same variable and exponent. This helps in reducing the complexity of an expression and makes it easier to work with.
To identify like terms, look for terms that contain the same variables raised to the same power. In our example, the terms \(10x\) and \(6x\) are like terms because they both have the variable \(x\) to the same power. Similarly, the constant terms \(5\) and \(4\) are also considered like terms.

• Add coefficients of like terms. For example, \(10x + 6x\) simplifies to \(16x\).
• Combine constants together. Here, \(5 + 4\) equals \(9\).

Successfully combining like terms results in a simpler expression, such as \(16x + 9\) in this problem.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to distribute a multiplier across terms inside parentheses. It is particularly useful when handling subtraction problems involving polynomials.
In subtraction, the distributive property helps when we change the subtraction of a negative to the addition of a positive. In our exercise, we initially have: \((10x + 5) - (-6x - 4)\).

• Change subtraction of a polynomial to the addition of its opposite. Thus the expression becomes \(10x + 5 + 6x + 4\).
• Ensures terms inside the parentheses are accurately processed.

Using the distributive property simplifies the task of combining like terms, making it a powerful tool for solving polynomial expressions.

Algebraic Expressions

Algebraic expressions are a compact way to convey mathematical ideas using variables, numbers, and operation symbols. They are foundational to mastering any algebraic concept or solution.
To interpret an algebraic expression, it’s important to understand components such as coefficients (numeric parts), variables (letters representing numbers), and terms (products of coefficients and variables). In our exercise, \(10x + 5 - (-6x - 4)\) is an algebraic expression.

• Polynomials: Expressions consisting of variables and coefficients, such as \(10x + 5\) and \(-6x - 4\).
• Operations: Be mindful of addition, subtraction, and distribution throughout the process.

Understanding the structure and properties of algebraic expressions allows you to manipulate and solve them effectively, as seen when transforming this expression to \(16x + 9\) by following algebraic rules.