Problem 79 - Beginning and Intermediate Algebra

Question

Each of the polynomials below is a polynomial in two variables. Perform the indicated operation(s). $$\left(5 m+\frac{5}{6} n+\frac{1}{2}\right)+\left(-6 m+n-\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The simplified polynomial expression after performing the addition operation is: \(-m + \frac{11}{6}n + \frac{1}{4}\).

Step 1: Identifying the polynomials to add

First, let's identify the two polynomials to add: Polynomial 1: \(5m+\frac{5}{6}n+\frac{1}{2}\) Polynomial 2: \(-6m+n-\frac{3}{4}\)

Step 2: Combining like terms

Now, we will add the two polynomials by combining their like terms: The like terms for variable \(m\) are \(5m\) and \(-6m\). The like terms for variable \(n\) are \(\frac{5}{6}n\) and \(n\).

Step 3: Adding the coefficients of variable \(m\)

Add the coefficients (numbers in front of the variable) of the like terms for variable \(m\): \(5m - 6m = -1m \ or -m\)

Step 4: Adding the coefficients of variable \(n\)

Add the coefficients of the like terms for variable \(n\): \(\frac{5}{6}n + n = \frac{5}{6}n+\frac{6}{6}n = \frac{11}{6}n\)

Step 5: Adding the constants

Add the constants (if any) in the polynomials: \(\frac{1}{2}-\frac{3}{4}=\frac{1}{4}\)

Step 6: Final Expression

Combine all the results to form the simplified polynomial expression: \(-m + \frac{11}{6}n + \frac{1}{4}\)

Key Concepts

Adding Polynomials

When we talk about adding polynomials, we mean combining two or more polynomials into one by summing them together. Each polynomial might have different numbers and combinations of variables, but the process of adding is simplified by focusing on similar parts of the expression, called like terms. You need to align terms with the same variables, then adjust the coefficients—these are the numbers in front of the variables.
For example, if you have two polynomials, like in the exercise given, you will first list each expression. Then, you'll look at each pair of terms to determine which can be added. You treat the variables almost like a label. They help you know which parts of each polynomial to add together. This method ensures that all parts are considered in the final solution.

Like Terms

Like terms are a crucial concept when adding polynomials because they determine which terms can be combined. A polynomial term generally consists of a coefficient and one or more variables. Like terms have the same variables raised to the same power.

• Example: Terms like \(3x\) and \(-5x\) are like terms because they both have the variable \(x\).
• However, \(2x\) and \(3y\) are not like terms because they have different variables.

In the given exercise, when you see terms like \(5m\) and \(-6m\), these are like terms because they both have the variable \(m\).
You combine them by adding their coefficients so \(5m - 6m\) results in \(-m\). Similarly, you handle other variable terms by adding the fractions after converting them to a common denominator, if necessary.

Variables in Polynomials

Variables in polynomials act like placeholders, representing numbers in equations and expressions. They are typically denoted by letters such as \(m\), \(n\), \(x\), or \(y\). These variables can be raised to various powers, though in the example given, they are all raised to the same power: power of one. Variable terms are fundamental in shaping polynomials. They allow polynomials to represent general rules of change and dynamic relationships between quantities. Variables are also what remain unchanged when you’re combining terms—only the coefficients in front of them are added or subtracted. In the exercise, recognizing the variables helps us group terms together during addition.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. After combining all like terms in the polynomial, you'll end up with an expression that is easier to understand and work with. This was demonstrated in our original problem, where all like terms were added:

• The \(m\) terms are combined to become \(-m\).
• The \(n\) terms are summed as \(\frac{11}{6}n\).
• The constant terms result in \(\frac{1}{4}\).

After this process, you have a simplified version of your original polynomial. Simplifying ensures that you have one clear expression to represent the entire relationship, making it easier to analyze or use in further calculations. Simplified expressions are preferred in mathematics for their clarity and utility.