Problem 19 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$(5 x+6)+(-2 x+6)$$

Step-by-Step Solution

Answer

The sum of the polynomials is \(3x + 12\).

Step 1: Identify Like Terms

Break down both polynomials into their individual terms. The first polynomial is \(5x + 6\), and the second is \(-2x + 6\). The like terms are those that have the same variable raised to the same power.

Step 2: Add the Variable Terms

Identify the terms with the variable \(x\). In this case, it's \(5x\) from the first polynomial and \(-2x\) from the second. Add these together: \(5x + (-2x) = 3x\).

Step 3: Add the Constant Terms

Next, add the constant terms from each polynomial. These terms are \(6\) from the first polynomial and \(6\) from the second polynomial. Add them together: \(6 + 6 = 12\).

Step 4: Combine the Results

Integrate the results from the variable terms and the constant terms: \(3x + 12\). This is the result of adding the polynomials.

Key Concepts

Like Terms

When dealing with polynomials, it's crucial to identify like terms. These are terms that have the same variable parts raised to the same power. For example, in the expression \(5x + 6 + (-2x + 6)\), the terms \(5x\) and \(-2x\) are like terms because they both involve the variable \(x\) raised to the first power. Similarly, the constants \(6\) and \(6\) are also like terms because they do not involve any variables.

• Like terms are combined by adding or subtracting their coefficients.
• If the terms involve a variable, ensure the variables match exactly.
• Recognizing like terms is the first essential step in simplifying polynomials.

By grouping and simplifying like terms, polynomials can be made more manageable, leading to simpler expressions.

Variable Terms

Variable terms are parts of an algebraic expression that include variables, such as \(x\). In polynomial addition, you focus on these terms separately from constants.
By isolating the variable terms, you simplify the process of polynomial addition. For the polynomial \(5x + 6 + (-2x + 6)\), the variable terms are \(5x\) and \(-2x\).

• Variable terms are like terms if they have the same variable raised to the same power.
• In practice, this means you can only add or subtract terms that are "like."
• For the example, combining \(5x\) and \(-2x\) yields \(3x\).

Keeping variable terms organized helps to quickly see which parts of an expression you can combine.

Constant Terms

Constant terms in polynomials are the numbers without any attached variables. They remain unchanged as they do not depend on any variable's value. In our example, both \(+6\) from the first polynomial and \(+6\) from the second polynomial are constant terms.
The process of dealing with constant terms parallels that for variable terms. Simply add the constant numbers together as you see them. In our given exercise, we add \(6 + 6\), resulting in \(12\).

• Constant terms are always added directly, as they do not include variables.
• Adding these terms is the second critical step after combining variable terms.
• Properly dealing with constant terms means ensuring they reflect the actual sum of the constants in the given expressions.

Understanding and handling constant terms correctly ensures the accuracy of your polynomial addition.