Problem 9 - Elementary Algebra

Question

For Problems \(9-22\), add the polynomials. $$ 3 x+4 \text { and } 5 x+7 $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(8x + 11\).

Step 1: Identify the like terms

In the expression given, namely the sum of polynomials \(3x + 4\) and \(5x + 7\), we need to identify the like terms. The terms containing \(x\) are like terms, and the constant terms (those without \(x\)) are also like terms.

Step 2: Add the like terms

Add the coefficients of the like terms. The terms containing \(x\) are \(3x\) and \(5x\). Add them together: \(3x + 5x = 8x\). The constant terms are \(4\) and \(7\); add them together: \(4 + 7 = 11\).

Step 3: Form the final polynomial

Combine the sums from the like terms: \(8x + 11\). This combination is the result of adding both polynomials \(3x + 4\) and \(5x + 7\).

Key Concepts

Understanding Like Terms

When working with polynomials, one of the first steps in simplifying or adding them is to identify like terms. Like terms are those that have the exact same variable raised to the same power. This means that the structure of the terms is identical, except possibly for their coefficients. For example, in the expression \(3x + 4\) and \(5x + 7\), the terms \(3x\) and \(5x\) are like terms because they both involve the variable \(x\). Similarly, the numbers \(4\) and \(7\) are also like terms, but they are known as constant terms, which we'll delve into later. Recognizing like terms is crucial because it dictates what terms can be combined directly in operations like addition or subtraction, leading to a simpler expression.

The Role of Coefficients in Polynomial Addition

Coefficients are the numerical part of terms that include variables in polynomials. For example, in the term \(3x\), the number \(3\) is the coefficient. Similarly, in \(5x\), the coefficient is \(5\). In polynomial addition, coefficients of like terms are added together. For the problem at hand, adding the polynomials \(3x + 4\) and \(5x + 7\), we add the coefficients of \(x\):

• For \(3x + 5x\), sum the coefficients \(3 + 5\) to get \(8\).

This step reduces the expression: the coefficients tell us how many of a particular like term we have to add, simplifying the overall polynomial expression into \(8x + 11\). Thus, mastering coefficients is essentially about mastering basic arithmetic with the numbers that multiply the variables in your expressions.

Unpacking Constant Terms

Constant terms are those terms in a polynomial that do not have any variables attached to them. In the expression \(3x + 4\) and \(5x + 7\), the constant terms are \(4\) and \(7\). Even though these terms do not contain variables, they are still part of the polynomial and contribute to the final result of operations. When adding polynomials, you combine the constant terms just as you would any other numbers:

• Add \(4 + 7\) to get \(11\).

In the resulting polynomial \(8x + 11\), the number \(11\) is the constant term. Understanding constant terms is important because it helps you complete the formation of the simplest version of an expression.