Let's perform the binomial multiplication to determine the values of the variables in the given table.
First, let's replicate the table for better understanding:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline & $x$ & 7 \\
\hline $2 x$ & $2 x^2$ & $B$ \\
\hline-3 & $A$ & $C$ \\
\hline
\end{tabular}
\][/tex]
Now, let's fill in the missing values in the table step-by-step:
1. Determining [tex]\( A \)[/tex]:
[tex]\( A \)[/tex] is the product of the term in its row and column. In the row where [tex]\( A \)[/tex] is located, the term is [tex]\(-3\)[/tex], and in its column, the term is [tex]\(x\)[/tex]:
[tex]\[
A = (-3) \cdot x
\][/tex]
Since [tex]\( x \)[/tex] is equal to [tex]\( 1 \)[/tex] (considering unit coefficient),
[tex]\[
A = -3
\][/tex]
2. Determining [tex]\( B \)[/tex]:
[tex]\( B \)[/tex] is the product of the term in its row and column. In the row where [tex]\( B \)[/tex] is located, the term is [tex]\(2x\)[/tex], and in its column, the term is [tex]\(7\)[/tex]:
[tex]\[
B = 2x \cdot 7
\][/tex]
Since [tex]\( x \)[/tex] is equal to [tex]\( 1 \)[/tex] (considering unit coefficient again),
[tex]\[
B = 2 \cdot 7
\][/tex]
[tex]\[
B = 14
\][/tex]
3. Determining [tex]\( C \)[/tex]:
[tex]\( C \)[/tex] is the product of the term in its row and column. In the row where [tex]\( C \)[/tex] is located, the term is [tex]\(-3\)[/tex], and in its column, the term is [tex]\(7\)[/tex]:
[tex]\[
C = -3 \cdot 7
\][/tex]
[tex]\[
C = -21
\][/tex]
Finally, we will identify the like terms from the table. Like terms are those that contain the same variables raised to the same power. In this table, the like terms are:
[tex]\[
A = -3x \quad \text{and} \quad B = 14x
\][/tex]
Both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] contain the variable [tex]\( x \)[/tex].
The values are:
[tex]\[ A = -3 \][/tex]
[tex]\[ B = 14 \][/tex]
[tex]\[ C = -21 \][/tex]
The letters that represent like terms in the table are [tex]\( A \)[/tex] and [tex]\( B \)[/tex], since they both include the variable [tex]\( x \)[/tex].
