- Identify the groups of like terms: \(3s^2\) and \(-s^2\), \(11s\) and \(-5s\), and \(-4\) and \(7\).
- Combine the squared terms: \(3s^2 - s^2 = 2s^2\).
- Combine the linear terms: \(11s - 5s = 6s\).
- Combine the constant terms: \(-4 + 7 = 3\).
- The final simplified expression is \(\boxed{2s^2 + 6s + 3}\).
Explanation
- Identifying Like Terms
We are given the algebraic expression \(3s^2 + 11s - s^2 - 4 - 5s + 7\) and asked to simplify it. To simplify an expression, we need to combine its 'like terms'. Like terms are terms that contain the same variable raised to the same power. In our case, we can observe the following sets of like terms:
- Squared terms involving \(s^2\): \(3s^2\) and \(-s^2\).
- Linear terms involving \(s\): \(11s\) and \(-5s\).
- Constant terms: \(-4\) and \(7\).
- Combining Like Terms
Now that we have grouped the like terms, we can combine them by adding or subtracting their coefficients:
- For the \(s^2\) terms: \(3s^2 - s^2 = (3 - 1)s^2 = 2s^2\).
- For the \(s\) terms: \(11s - 5s = (11 - 5)s = 6s\).
- For the constant terms: \(-4 + 7 = 3\).
- Final Simplified Expression
By summing the results obtained from combining each group of like terms, we construct the simplified version of the original expression:
\[2s^2 + 6s + 3.\]
Comparing this result to the given options (\(8s^2 + 3\), \(8s + 3\), \(2s^2 + 6s\), \(2s^2 + 6s + 3\)), we identify that the simplified expression is \(2s^2 + 6s + 3\).
Examples
Imagine you are stocking shelves in a store. You have different items: some large boxes (\(s^2\)), some medium-sized bags (\(s\)), and some small individual items (constants). If you start with 3 boxes, 11 bags, and lose 4 items, then realize you need to remove 1 box, remove 5 bags, and add 7 items, you are effectively tracking your inventory change. Simplifying algebraic expressions is exactly like tallying these different categories of items so you know your final stock level of each type without getting confused by the jumble of additions and subtractions.