Problem 80 - Beginning and Intermediate Algebra

Question

Each of the polynomials below is a polynomial in two variables. Perform the indicated operation(s). $$\left(-2 c-\frac{2}{3} d+1\right)-\left(2 c+\frac{1}{9} d-\frac{4}{7}\right)$$

Step-by-Step Solution

Answer

The short answer is: \(P_1 - P_2 = -4c -\frac{7}{9}d + \frac{11}{7}\)

Step 1: Write down the given polynomials

We are given the following polynomials: \( P_1 = -2c - \frac{2}{3}d + 1 \) \( P_2 = 2c + \frac{1}{9}d - \frac{4}{7} \)

Step 2: Subtract the second polynomial from the first polynomial

To subtract the second polynomial from the first one, we need to subtract the corresponding terms of the second polynomial from the first one: $$P_1 - P_2 = (-2c - \frac{2}{3}d + 1) - (2c + \frac{1}{9}d - \frac{4}{7})$$

Step 3: Distribute the negative sign across the second polynomial and combine like terms

Now, we distribute the negative sign across the second polynomial and combine like terms: $$P_1 - P_2 = -2c - \frac{2}{3}d + 1 - 2c - \frac{1}{9}d + \frac{4}{7}$$

Step 4: Combine the coefficients of the variables and the constant terms

We'll now combine the coefficients of "c", "d" and the constant terms separately: For "c": \(-2c - 2c = -4c \) For "d": \(-\frac{2}{3}d - \frac{1}{9}d = -\frac{7}{9}d \) For the constants: \(1 + \frac{4}{7} = \frac{11}{7} \)

Step 5: Write down the final result

Now, we'll write down the final result of the subtraction: $$P_1 - P_2 = -4c -\frac{7}{9}d + \frac{11}{7}$$

Key Concepts

Subtracting Polynomials

Subtracting polynomials involves finding the difference between two polynomial expressions. The key to this operation is understanding the role of the minus sign. When subtracting one polynomial from another, it is important to subtract each corresponding term. This means you will perform subtraction for each group of like terms separately.

Here are steps to effectively subtract polynomials:

• Write down both polynomials: Clearly identify the polynomials that need to be subtracted. Each polynomial consists of terms including coefficients, variables, and powers.
• Apply the negative sign: Distribute the negative sign across the second polynomial. This will change the sign of every term in the second polynomial.
• Subtract corresponding terms: Subtract terms with the same variables and exponents from each other. This involves simple arithmetic operations on the coefficients.
• Combine results: Once you have subtracted all the corresponding terms, combine the results to get the final simplified expression.

By carefully applying these steps, you can subtract any set of polynomials, resulting in another polynomial that expresses the difference between them.

Two Variables

Polynomials can consist of multiple variables. Each variable may be raised to different powers and may have different coefficients attached. In this exercise, we are dealing with polynomials in two variables: \(c\) and \(d\).

To work with polynomials with more than one variable:

• Identify Like Terms: Look for terms that have the same variables raised to the same power. For example, \(-2c\) and \(2c\) are like terms because they share the same variable and exponent.
• Align Variables: It’s useful to organize your terms so that like terms are lined up in the same position for easy comparison and combination.
• Operate Separately: Perform operations on each variable group separately. This helps avoid errors and makes combining terms simpler.

Working with polynomials with two variables requires systematically breaking down the expression into parts that involve each variable separately.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have the same variables and exponents. This is an essential step in simplifying polynomials and performing arithmetic operations like addition and subtraction.

Here's how you can effectively combine like terms:

• Identify like terms: Look for terms in the expression that use the same variable and have the same exponent. Only these can be combined.
• Add or subtract coefficients: Once you have identified like terms, focus just on the coefficients – the numerical parts – and add or subtract these as required by the operation.
• Recombine the terms: Attach the combined coefficient to the variable part of the terms to form a new term.

By following these steps, the polynomial becomes much simpler. In subtraction, subtracting similar terms one by one and then combining them creates a streamlined, manageable polynomial that is much easier to understand and use.