Sure! Let's work through the binomial multiplication step-by-step to determine the values of the letters [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
The binomial multiplication represented is [tex]\((x + 7)(2x - 3)\)[/tex]. We need to determine the values in the resulting multiplication table.
[tex]\[
\begin{array}{|c|c|c|}
\hline
& x & 7 \\
\hline
2x & 2x^2 & B \\
\hline
-3 & A & C \\
\hline
\end{array}
\][/tex]
Let's fill in each cell:
1. Top left cell: [tex]\(2x \times x = 2x^2\)[/tex] (This is already given)
2. Top right cell (B): [tex]\(2x \times 7 = 14x\)[/tex]
3. Bottom left cell (A): [tex]\(-3 \times x = -3x\)[/tex]
4. Bottom right cell (C): [tex]\(-3 \times 7 = -21\)[/tex]
Now we have filled in the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
& x & 7 \\
\hline
2x & 2x^2 & 14x \\
\hline
-3 & -3x & -21 \\
\hline
\end{array}
\][/tex]
Hence, the values of the letters in the table are:
[tex]\[
A = -3x, \quad B = 14x, \quad C = -21
\][/tex]
To identify like terms:
- Like terms are terms that contain the same variable raised to the same power.
- In this case, [tex]\(A = -3x\)[/tex] and [tex]\(B = 14x\)[/tex] are like terms because they both contain the variable [tex]\(x\)[/tex] raised to the first power.
So, the letters from the table that represent like terms are [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Thus, the solution is:
[tex]\[
A = -3x, \quad B = 14x, \quad C = -21
\][/tex]
The like terms are [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
