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Add $\left(1.3 t^3+0.4 t^2-24 t\right)+\left(8-18 t+0.6 t^2\right)$



For each term in the second polynomial, select the letter showing where that term should be placed to add the polynomials vertically.



$\begin{array}{l}

1.3 t^3+0.4 t^2+(-24 t) \\

+\quad A \quad B \quad C \quad D \\

\hline

\end{array}$



The term $0.6 t^2$ should be placed at [ ].



The term $0 t^3$ should be placed at [ ].



The term 8 should be placed at [ ].



The term $-18 t$ should be placed at [ ].

Verified Answer :

  • Determine the column degrees based on the first polynomial: A = \(t^3\), B = \(t^2\), C = \(t^1\), and D = constant (\(t^0\)).
  • Align each term of the second polynomial \(P_2 = 0.6t^2 - 18t + 8\) (and \(0t^3\)) with its corresponding column.
  • The term \(0.6t^2\) belongs in column B.
  • The term \(0t^3\) belongs in column A.
  • The term \(8\) belongs in column D.
  • The term \(-18t\) belongs in column C.
  • Final placement summary: \(\boxed{0.6t^2 \rightarrow B, 0t^3 \rightarrow A, 8 \rightarrow D, -18t \rightarrow C}\).

Explanation

  1. Analyzing the Polynomial Structure and Column Alignment

We are given two polynomials: \(P_1 = 1.3t^3 + 0.4t^2 - 24t\) and \(P_2 = 8 - 18t + 0.6t^2\). We aim to add these polynomials by aligning them vertically based on the power of the variable \(t\). The vertical arrangement is provided as:
\[ \begin{array}{l} 1.3 t^3 + 0.4 t^2 + (-24 t) \ +\quad A \quad B \quad C \quad D \\ \end{array} \]
Each column corresponds to a specific degree of the variable \(t\). We can identify the columns by observing the first polynomial:

  • The first term \(1.3t^3\) is at the \(t^3\) position, implying column \(A\) is the \(t^3\) column.
  • The second term \(0.4t^2\) is at the \(t^2\) position, implying column \(B\) is the \(t^2\) column.
  • The third term \(-24t\) is at the \(t^1\) position, implying column \(C\) is the \(t^1\) column.
  • Since there is no constant term in \(P_1\), column \(D\) represents the constant terms, or \(t^0\).
  1. Aligning the Terms of the Second Polynomial

Now we categorize the terms of the second polynomial \(P_2 = 0.6t^2 - 18t + 8\) to align them with the established columns \(A, B, C, D\):

  • The term \(0.6t^2\) has a degree of 2, so it belongs in the \(t^2\) column, which is column \(B\).
  • The term \(0t^3\) represents the missing degree-3 term, which belongs in the \(t^3\) column, column \(A\).
  • The term \(8\) is a constant, so it belongs in the constant term column, which is column \(D\).
  • The term \(-18t\) has a degree of 1, so it belongs in the \(t^1\) column, which is column \(C\).
  1. Assigning Terms to Columns

Based on our analysis, we assign each term to its respective lettered column:

  • The term \(0.6t^2\) should be placed at \(B\).
  • The term \(0t^3\) should be placed at \(A\).
  • The term \(8\) should be placed at \(D\).
  • The term \(-18t\) should be placed at \(C\).

Examples

Polynomial addition is a fundamental operation used in various fields, such as engineering and physics, to combine different components of a system. For example, if you are modeling the trajectory of a projectile where different forces act at different times or represented by different polynomials, you can sum these models by aligning terms of the same power. This systematic approach ensures that you only combine like terms (e.g., \(t^2\) with \(t^2\)), which is essential for accurate modeling and solving real-world design problems.

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