Answered

Find the sum.



$\left(3 n^2-5 n+6

ight)+\left(-8 n^2-3 n-2

ight)=$


A$-5 n^2-8 n+4$
B$-11 n^2-2 n+4$
C$-5 n^2-2 n-4$
D$-11 n^2-8 n-4$

Verified Answer :

  • Identify and group the like terms for \(n^2\), \(n\), and the constants from the two polynomials.
  • Perform the addition for each group: \((3-8)n^2 = -5n^2\), \((-5-3)n = -8n\), and \(6-2 = 4\).
  • Combine these results to obtain the final polynomial: \(-5n^2 - 8n + 4\).
  • The final answer corresponds to option a: \(\boxed{-5n^2 - 8n + 4}\).

Explanation

  1. Analyzing the Problem

To find the sum of the two given quadratic polynomials, \((3n^2 - 5n + 6)\) and \((-8n^2 - 3n - 2)\), we need to combine the like terms. Like terms are terms that have the same variables raised to the same power. In this expression, we have terms with \(n^2\), terms with \(n\), and constant terms. We can add these polynomials by grouping these corresponding terms together.

  1. Combining Like Terms

First, we group the coefficients of the \(n^2\) terms:
\[3n^2 + (-8n^2) = (3 - 8)n^2 = -5n^2\]
Next, we group the coefficients of the \(n\) terms:
\[-5n + (-3n) = (-5 - 3)n = -8n\]
Finally, we add the constant terms:
\[6 + (-2) = 4\]

  1. Final Summation and Verification

By combining the results from the previous step, we form the resulting quadratic polynomial:
\[-5n^2 - 8n + 4\]
Comparing this result to the provided options:

A\(-5n^2 - 8n + 4\)
B\(-11n^2 - 2n + 4\)
C\(-5n^2 - 2n - 4\)
D\(-11n^2 - 8n + 4\)

The sum is equivalent to option (a).

Examples

Polynomial addition is a fundamental skill in algebra used in various fields, such as engineering and physics. For example, if two forces acting on an object are represented by quadratic functions of time, you would add these functions to find the net force acting on the object at any given time. Understanding how to group and combine terms allows you to simplify complex expressions into manageable forms for easier analysis.

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