Problem 33 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ (0.3 p+2.1 q)+(0.4 p-3 q) $$

Step-by-Step Solution

Answer

The sum is \( 0.7p - 0.9q \).

Step 1: Identify Like Terms

In the given problem, identify the like terms from each polynomial:\( 0.3p \) and \( 0.4p \) are like terms. Similarly, \( 2.1q \) and \( -3q \) are also like terms.

Step 2: Add the Coefficients of Like Terms

Add the coefficients of the like terms separately. For the \( p \) terms, calculate \( 0.3 + 0.4 = 0.7p \). For the \( q \) terms, calculate \( 2.1 + (-3) = -0.9q \).

Step 3: Combine the Results

Combine the results from Step 2 to form the final polynomial: \( 0.7p - 0.9q \).

Key Concepts

Like Terms

**Understanding Like Terms in Polynomials**
In polynomial addition, like terms play a critical role. These are terms within polynomials that possess the same variable raised to the same power. In simpler words, they must look identical, apart from the number (coefficient) in front. For example, in the expression \( 0.3p + 2.1q + 0.4p - 3q \), terms such as \( 0.3p \) and \( 0.4p \) are like terms because they both have the variable \( p \). Similarly, \( 2.1q \) and \( -3q \) match each other as like terms due to the \( q \) variable.

Identifying like terms is essential because it allows us to perform addition on these terms. It is crucial to ensure that only terms with the same variables and exponents are combined during the addition process. This is the reason why terms like \( p \) and \( q \) are not considered like terms as they have different variables.

To effectively work with polynomials, mastering the identification of like terms will simplify the process of combining and solving them.

Coefficients

**The Role of Coefficients in Polynomials**
In any polynomial, each term consists of a variable and a coefficient. Coefficients are the numbers placed in front of the variables, acting as multipliers. For instance, in the term \( 0.3p \), \( 0.3 \) is the coefficient.

Some key things to remember about coefficients:

• Coefficients can be any real number – positive, negative, or zero.
• When adding like terms, the coefficients are what you combine, while the variable part remains unchanged.

In our exercise, after identifying like terms \( 0.3p \) and \( 0.4p \), we add their coefficients together: \( 0.3 + 0.4 = 0.7 \). This results in the term \( 0.7p \). Similarly, you sum up the coefficients of \( 2.1q \) and \( -3q \) to get \( -0.9q \).

These coefficients are vital as they directly affect the value of each term and, consequently, the entire polynomial. Understanding the role of coefficients aids in effectively simplifying and combining polynomials.

Combining Polynomials

**The Process of Combining Polynomials**
Combining polynomials involves the straightforward process of finding the sum of these expressions. This procedure is predominantly about combining like terms and simplifying the polynomial as thoroughly as possible.

Here's a step-by-step outline based on our task:

• Identify the like terms within each polynomial. For our exercise, this included \( 0.3p \) with \( 0.4p \) and \( 2.1q \) with \( -3q \).
• Add the coefficients of these like terms. For \( p \): \( 0.3 + 0.4 = 0.7p \), and for \( q \): \( 2.1 + (-3) = -0.9q \). This means we have simplified the problem into \( 0.7p - 0.9q \).
• Form the final polynomial by combining these simplified terms: \( 0.7p - 0.9q \).

By following these steps, you ensure that the addition of polynomials is both efficient and accurate. With practice, combining polynomials can become a fluid and intuitive process, vastly simplifying algebraic operations.