To add the two polynomials [tex]\(\left(1.3 t^3 + 0.4 t^2 - 24 t \right) + \left(8 - 18 t + 0.6 t^2 \right)\)[/tex], we need to align like terms and add their coefficients.
Here's a step-by-step solution:
1. Identify and align the terms of the polynomials based on their degrees.
- The terms in the first polynomial are:
[tex]\[
\begin{align*}
1.3 t^3 & \quad (\text{coefficient is } 1.3) \\
0.4 t^2 & \quad (\text{coefficient is } 0.4) \\
-24 t & \quad (\text{coefficient is } -24) \\
0 & \quad (\text{constant term})
\end{align*}
\][/tex]
- The terms in the second polynomial are:
[tex]\[
\begin{align*}
0 t^3 & \quad (\text{coefficient is } 0) \\
0.6 t^2 & \quad (\text{coefficient is } 0.6) \\
-18 t & \quad (\text{coefficient is } -18) \\
8 & \quad (\text{constant term})
\end{align*}
\][/tex]
2. Align the like terms:
[tex]\[
\begin{array}{ccc}
& 1.3 t^3 + & 0 t^3 \\
& 0.4 t^2 + & 0.6 t^2 \\
& -24 t + & -18 t \\
& 0 + & 8
\end{array}
\][/tex]
3. Add the coefficients of the like terms:
[tex]\[
\begin{align*}
\text{Coefficient for } t^3 : & \quad 1.3 + 0 = 1.3 \\
\text{Coefficient for } t^2 : & \quad 0.4 + 0.6 = 1.0 \\
\text{Coefficient for } t : & \quad -24 + (-18) = -42 \\
\text{Constant term} : & \quad 0 + 8 = 8
\end{align*}
\][/tex]
4. Write the resulting polynomial:
[tex]\[
1.3 t^3 + 1.0 t^2 - 42 t + 8
\][/tex]
Hence, when we add the polynomials [tex]\(\left(1.3 t^3 + 0.4 t^2 - 24 t \right) + \left(8 - 18 t + 0.6 t^2 \right)\)[/tex], the result is:
[tex]\[
1.3 t^3 + 1.0 t^2 - 42 t + 8
\][/tex]
Corresponding letters for where the terms should be placed:
- [tex]\(\boxed{0.6 t^2 \text{ should be placed aligned with the } 0.4 t^2 \text{ term}}\)[/tex]
- [tex]\(\boxed{0 t^3 \text{ should be placed aligned with the } 1.3 t^3 \text{ term} (or just omitted as it is zero)}\)[/tex]
- [tex]\(\boxed{8 \text{ should be placed in the constant term position}}\)[/tex]
- [tex]\(\boxed{-18 t \text{ should be placed aligned with the } -24 t \text{ term}}\)[/tex]
