Answered

Choose the correct values for A, B, C, and D that align like terms to find the sum vertically.



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




\multicolumn{7}{|c|}{$1.3 t^3+0.4 t^2+(-24 t)$}

+		A +B+C+D





A$A = 0 t^3, B = (-0.6 t^2), C = 18 t, D = (-8)$
B$A = 0 t^3, B = (-0.6 t^2), C = (-8), D = 18 t$
C$A = (-0.6 t^2), B = (-8), C = 18 t, D = 0 t^3$

Verified Answer :

  • Analyze the degree order of the given polynomial: \(t^3, t^2, t^1, t^0\).
  • Align the terms of the second polynomial \(A, B, C, D\) by matching these degrees respectively: \(A\) for \(t^3\), \(B\) for \(t^2\), \(C\) for \(t^1\), and \(D\) for \(t^0\).
  • Compare the options and identify that option 1 maintains the correct degree-based order.
  • The final answer is \(\boxed{1}\).

Explanation

  1. Analyzing the Problem and Polynomial Alignment

We are presented with a polynomial expression \(1.3 t^3 + 0.4 t^2 - 24 t\) and need to add a second polynomial represented by \(A + B + C + D\). The task is to identify which of the three given sets of \(A, B, C, D\) allows for a logical "vertical" alignment of like terms.

Vertical alignment of polynomials requires that terms of the same degree (i.e., the same power of \(t\)) are grouped together. The original polynomial is \(1.3 t^3 + 0.4 t^2 - 24 t\), which has terms of degree 3, 2, and 1. The constant term is 0. Adding a polynomial \(A + B + C + D\) implies that these four components will correspond to the columns of degrees 3, 2, 1, and 0 respectively, based on the provided table structure:

\multicolumn{7}{|c|}{\(1.3 t^3 + 0.4 t^2 + (-24 t)\)} \) +\(A\)+\(B\)+\(C\)+\(D\) \)

Wait, the provided table structure in the problem description looks slightly different, but the goal remains to match like terms. Let's analyze the standard polynomial addition format. A vertical sum typically aligns:

  • \(t^3\) column
  • \(t^2\) column
  • \(t^1\) column
  • \(t^0\) (constant) column

The expression is \(1.3 t^3 + 0.4 t^2 - 24 t + 0\). We seek a set \(\{A, B, C, D\}\) such that when aligned with the original, the powers of \(t\) match consistently. Let's look at the degrees present in the options:
Option 1: \(A = 0 t^3, B = -0.6 t^2, C = 18 t, D = -8\)
Option 2: \(A = 0 t^3, B = -0.6 t^2, C = -8, D = 18 t\)
Option 3: \(A = -0.6 t^2, B = -8, C = 18 t, D = 0 t^3\)

In the standard left-to-right representation \(1.3 t^3 + 0.4 t^2 - 24 t + 0\), the columns follow the sequence of degrees \(3, 2, 1, 0\).
Therefore, we require:

  • Column 1 (\(t^3\)): \(A\)
  • Column 2 (\(t^2\)): \(B\)
  • Column 3 (\(t^1\)): \(C\)
  • Column 4 (\(t^0\)): \(D\)

Matching this to the options:
Option 1: \(A=0t^3\), \(B=-0.6t^2\), \(C=18t\), \(D=-8\). This matches the sequence of degrees 3, 2, 1, 0.
This makes sense as it maintains the degree order.

  1. Evaluating the Options

Comparing the requirements with the given options:

  • Original expression: \(1.3 t^3 + 0.4 t^2 - 24 t + 0\)
  • Structure: \(A\) (degree 3) + \(B\) (degree 2) + \(C\) (degree 1) + \(D\) (constant)

Checking Option 1:

  • \(A = 0 t^3\) (degree 3)
  • \(B = -0.6 t^2\) (degree 2)
  • \(C = 18 t\) (degree 1)
  • \(D = -8\) (degree 0)

This alignment is correct as it places terms of the same degree in their respective columns, corresponding to the degrees 3, 2, 1, and 0.

Checking Option 2:

  • \(A = 0 t^3\) (degree 3)
  • \(B = -0.6 t^2\) (degree 2)
  • \(C = -8\) (degree 0)
  • \(D = 18 t\) (degree 1)

Here, the degrees are 3, 2, 0, 1. This violates the orderly column structure.

Checking Option 3:

  • \(A = -0.6 t^2\) (degree 2)
  • \(B = -8\) (degree 0)
  • \(C = 18 t\) (degree 1)
  • \(D = 0 t^3\) (degree 3)

The degrees here are 2, 0, 1, 3. This also violates the structure.

Thus, Option 1 is the only correct set of values that aligns like terms vertically according to their powers of \(t\).

  1. Conclusion

The correct assignment of values to align like terms vertically with \(1.3 t^3 + 0.4 t^2 - 24 t + 0\) is given by option 1.
The resulting sum is:
\((1.3 t^3 + 0 t^3) + (0.4 t^2 - 0.6 t^2) + (-24 t + 18 t) + (0 - 8) = 1.3 t^3 - 0.2 t^2 - 6 t - 8\).
The choice is indeed 1.

Examples

In computer science and data processing, algorithms often need to perform operations on arrays representing polynomials. If you have two polynomials, represented as vectors of coefficients like \([1.3, 0.4, -24, 0]\) and \([0, -0.6, 18, -8]\), adding these together requires aligning the coefficients by their corresponding power of \(t\). This is exactly what we just did: index 0 is \(t^3\), index 1 is \(t^2\), index 2 is \(t^1\), and index 3 is \(t^0\). This simple alignment ensures the addition is performed correctly, which is fundamental for systems that process algebraic inputs.