- Identify the terms for each variable and constants in the given polynomials: \(17m - 12n - 1\) and \(4 - 13m - 12n\).
- Group like terms: \((17 - 13)m\), \((-12 - 12)n\), and \((-1 + 4)\).
- Perform the arithmetic to simplify: \(4m - 24n + 3\).
- Match the result with the given options, identifying d as the correct choice: \(\boxed{d}\).
Explanation
- Analyze the polynomials to be summed
To find the sum of the two given polynomials, we need to add them together by combining their like terms. The two polynomials are \(17m - 12n - 1\) and \(4 - 13m - 12n\). Summing these involves grouping the terms containing \(m\), the terms containing \(n\), and the constant terms separately.
- Combine like terms
We identify the coefficients for each variable and the constants:
- For \(m\): \(17m\) and \(-13m\). The sum is \(17m - 13m = (17 - 13)m = 4m\).
- For \(n\): \(-12n\) and \(-12n\). The sum is \(-12n - 12n = (-12 - 12)n = -24n\).
- For the constants: \(-1\) and \(4\). The sum is \(-1 + 4 = 3\).
- Final result and comparison
By combining the results from the previous step, the sum of the two polynomials is:
\(4m - 24n + 3\).
Comparing this with the given options:
A\(4m + 3\)
B\(30m - 24n - 5\)
C\(30m - 5\)
D\(4m - 24n + 3\)
The calculated sum matches option d.
Examples
Polynomial addition is essential in various fields like physics and engineering, where you might need to combine multiple force vectors or signal components defined by variable expressions. For example, if you are modeling the total cost of a project as the sum of material costs and labor costs, each represented as a polynomial, you would group the like variables (like number of units or hours) to find the simplified total cost equation. Understanding how to aggregate these terms helps in simplifying complex systems into manageable mathematical models.