- Distribute the negative sign to all terms of the second polynomial to get \(-6 s^2 + 12 s - 8 - 3 s^2 - 8 s + 6\).
- Group like terms by powers of \(s\) to get \((-6 s^2 - 3 s^2) + (12 s - 8 s) + (-8 + 6)\).
- Perform the arithmetic to simplify to \(-9 s^2 + 4 s - 2\).
- The correct choice is option b, which is \(\boxed{-9 s^2 + 4 s - 2}\).
Explanation
- Analysis of the Polynomial Subtraction Problem
We are tasked with finding the difference between two quadratic polynomials: \((-6 s^2 + 12 s - 8)\) and \((3 s^2 + 8 s - 6)\). Our objective is to subtract the second polynomial from the first and simplify the expression to standard polynomial form. This involves distributing the subtraction sign across the terms of the second polynomial and then grouping and combining like terms.
- Distributing and Grouping Like Terms
To subtract the polynomials, we first remove the parentheses. The first set of parentheses remains unchanged, while the subtraction sign before the second set of parentheses changes the sign of each term inside:
\[ (-6 s^2 + 12 s - 8) - (3 s^2 + 8 s - 6) = -6 s^2 + 12 s - 8 - 3 s^2 - 8 s + 6 \]
Now, we group the like terms (terms with the same power of \(s\)):
\[ (-6 s^2 - 3 s^2) + (12 s - 8 s) + (-8 + 6) \]
- Simplifying the Polynomial
Now, we perform the arithmetic for each group:
- For the \(s^2\) terms: \(-6 s^2 - 3 s^2 = (-6 - 3) s^2 = -9 s^2\)
- For the \(s\) terms: \(12 s - 8 s = (12 - 8) s = 4 s\)
- For the constants: \(-8 + 6 = -2\)
Combining these results gives the simplified polynomial:
\[ -9 s^2 + 4 s - 2 \]
- Conclusion and Selection of the Correct Option
Comparing our result \(-9 s^2 + 4 s - 2\) with the given options:
A\(-9 s^2 + 20 s - 14\)
B\(-9 s^2 + 4 s - 2\)
C\(-9 s^2 + 4 s - 14\)
D\(-9 s^2 + 20 s - 2\)
Our result matches option b. Therefore, the difference is \(\boxed{-9 s^2 + 4 s - 2}\).
Examples
In real-world engineering or physics, we often deal with balancing equations or adjusting variables. Suppose you are modeling the velocity of two separate moving objects, where their positions are defined by quadratic functions. To find the relative difference in their displacement or velocity over time, you would perform a subtraction of these polynomial expressions, much like we did here. This allows analysts to determine the net behavior of a system when multiple forces are acting upon it simultaneously.