- Group the like terms of the polynomials by their exponent: \(1.3 t^3\), \((0.4 t^2 + 0.6 t^2)\), \((-24 t - 18 t)\), and the constant \(8\).
- Simplify the coefficients for each term to get \(1.3 t^3 + 1 t^2 - 42 t + 8\).
- The correct polynomial sum is \(\boxed{1.3 t^3 + t^2 - 42 t + 8}\), which corresponds to option d.
Explanation
- Problem Analysis
The objective is to find the sum of two given polynomials: \( (1.3 t^3 + 0.4 t^2 - 24 t) \) and \( (8 - 18 t + 0.6 t^2) \). To add polynomials, we need to combine the terms that have the same power of the variable \(t\). This process is known as grouping like terms. We will identify the coefficients for each power of \(t\) and perform the necessary additions.
- Calculating the Sum
The polynomials are:
\( P_1 = 1.3 t^3 + 0.4 t^2 - 24 t \)
\( P_2 = 8 - 18 t + 0.6 t^2 \)
We can write the sum as:
\( S = (1.3 t^3 + 0.4 t^2 - 24 t) + (0.6 t^2 - 18 t + 8) \)
Now, let's group the terms by the power of \(t\):
- For \(t^3\): \(1.3 t^3\)
- For \(t^2\): \(0.4 t^2 + 0.6 t^2 = (0.4 + 0.6) t^2 = 1.0 t^2 = t^2\)
- For \(t^1\): \(-24 t - 18 t = (-24 - 18) t = -42 t\)
- Constant term (\(t^0\)): \(8\)
Combining these results, the sum is:
\( S = 1.3 t^3 + t^2 - 42 t + 8 \)
- Comparison and Conclusion
Comparing our calculated result \(1.3 t^3 + t^2 - 42 t + 8\) with the given options:
A\(1.3 t^3 + t^2 - 6 t + 8\)
B\(1.9 t^2 - 42 t + 8\)
C\(1.9 t^3 + 8.4 t^2 - 42 t\)
D\(1.3 t^3 + t^2 - 42 t + 8\)
Our result matches option d. Thus, the correct answer is option d. The final expression is \(\boxed{1.3 t^3 + t^2 - 42 t + 8}\).
Examples
Adding polynomials is frequently used in fields like physics and engineering to model complex systems. For instance, if you are modeling the trajectory of a projectile where the displacement is given by the sum of two different motion components (such as constant velocity and acceleration), you would sum the polynomials representing those individual components to find the final position equation at any time \(t\). This technique allows for simplifying complex models into a single, manageable expression.