- Group like terms by identifying the coefficients of \(m\), \(n\), and constants.
- Perform the additions: \(17m - 13m = 4m\), \(-12n - 12n = -24n\), and \(-1 + 4 = 3\).
- Combine these results to get the sum: \(4m - 24n + 3\).
- Select the matching option, which is c. \(\boxed{4m - 24n + 3}\).
Explanation
- Analyzing the Polynomial Addition
The problem asks to find the sum of two polynomials: \(17m - 12n - 1\) and \(-13m - 12n + 4\). To find the sum of these expressions, we must add the corresponding terms together: the coefficients of \(m\), the coefficients of \(n\), and the constant terms. Polynomial addition follows the commutative and associative properties, allowing us to group like terms.
- Grouping and Adding Like Terms
We group the coefficients of the \(m\) terms, the coefficients of the \(n\) terms, and the constant terms as follows:
- The \(m\) terms: \(17m + (-13m) = (17 - 13)m = 4m\).
- The \(n\) terms: \(-12n + (-12n) = (-12 - 12)n = -24n\).
- The constant terms: \(-1 + 4 = 3\).
- Final Result and Selection
By combining the results of these additions, we form the resultant polynomial: \(4m - 24n + 3\). Comparing this to the given options:
A\(30m - 5\)
B\(30m - 24n - 5\)
C\(4m - 24n + 3\)
D\(4m + 3\)
The result matches option c.
Examples
Polynomial addition is a fundamental tool in algebra used for modeling real-world situations, such as managing inventory or budgeting. For example, if you have two separate budget plans where one tracks costs of two different raw materials and the other combines them, adding the polynomials allows you to determine the total cost as a function of the quantities needed. This simplifies complex problems by grouping expenses together, making it easier to predict total costs based on different variables.