Let's walk through constructing the table row by row based on the final answer.
### Step 1: Define the Structure
We have a 3x3 table with 3 rows and 3 columns. Each cell can contain either a polynomial expression or be left empty (denoted as `None`).
### Step 2: Populating the First Row
The first row is essentially the header row, excluding the first cell which is empty. The entries are:
- The first cell is empty.
- The second cell contains [tex]\(3x^2\)[/tex].
- The third cell contains [tex]\(5x\)[/tex].
So, the first row will be:
[tex]\[ [None, "3x^2", "5x"] \][/tex]
### Step 3: Populating the Second Row
For the second row, the entries are:
- The first cell contains [tex]\(4x^3\)[/tex].
- The second cell contains [tex]\(12x^6\)[/tex].
- The third cell contains [tex]\(20x^4\)[/tex].
So, the second row will be:
[tex]\[ ["4x^3", "12x^6", "20x^4"] \][/tex]
### Step 4: Populating the Third Row
For the third row, the entries are:
- The first cell contains [tex]\(2x^2\)[/tex].
- The second cell contains [tex]\(6x^4\)[/tex].
- The third cell contains [tex]\(7x^3\)[/tex].
So, the third row will be:
[tex]\[ ["2x^2", "6x^4", "7x^3"] \][/tex]
### Step 5: Construct the Final Table
Putting all the rows together, the final structured table looks like this:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
& $3x^2$ & $5x$ \\
\hline
$4x^3$ & $12x^6$ & $20x^4$ \\
\hline
$2x^2$ & $6x^4$ & $7x^3$ \\
\hline
\end{tabular}
\][/tex]
This table accurately presents the data row by row as we have constructed and verified.
