Consider the binomial multiplication represented in this table.

\begin{tabular}{|c|c|c|}
\hline
& [tex]$x$[/tex] & 7 \\
\hline
[tex]$2x$[/tex] & [tex]$2x^2$[/tex] & B \\
\hline
-3 & A & C \\
\hline
\end{tabular}

Perform the binomial multiplication to determine the value of the letters in the table.

[tex]\[ A = \square \][/tex]
[tex]\[ B = \square \][/tex]
[tex]\[ C = \square \][/tex]

Which letters from the table represent like terms?



Answer :

To solve the binomial multiplication [tex]\((2x - 3)(x + 7)\)[/tex] and determine the value of the letters in the given table, let's break down the multiplication step-by-step.

1. Multiplying each term in [tex]\(2x - 3\)[/tex] by each term in [tex]\(x + 7\)[/tex]:

[tex]\[ (2x - 3)(x + 7) \][/tex]

This can be expanded as:

[tex]\[ = (2x) \cdot (x) + (2x) \cdot (7) + (-3) \cdot (x) + (-3) \cdot (7) \][/tex]

2. Calculate each part of the expansion:

[tex]\[ = 2x^2 \quad \text{(from } 2x \cdot x\text{)} \][/tex]

[tex]\[ = 14x \quad \text{(from } 2x \cdot 7\text{, which is B in the table)} \][/tex]

[tex]\[ = -3x \quad \text{(from } -3 \cdot x\text{, which is A in the table)} \][/tex]

[tex]\[ = -21 \quad \text{(from } -3 \cdot 7\text{, which is C in the table)} \][/tex]

Now we collect the terms from the table:

- [tex]$A = -3x$[/tex]
- [tex]$B = 14x$[/tex]
- [tex]$C = -21$[/tex]

To find which letters represent like terms, we look for terms with the same variable part.

- [tex]\(A = -3x\)[/tex] (contains [tex]\(x\)[/tex])
- [tex]\(B = 14x\)[/tex] (contains [tex]\(x\)[/tex])
- [tex]\(C = -21\)[/tex] (constant term with no [tex]\(x\)[/tex])

So, the like terms are [tex]$A$[/tex] and [tex]$B$[/tex] because they both contain the variable [tex]\(x\)[/tex].

### Summary:
- [tex]\(A = -3x\)[/tex]
- [tex]\(B = 14x\)[/tex]
- [tex]\(C = -21\)[/tex]
- The letters representing like terms are [tex]\(A\)[/tex] and [tex]\(B\)[/tex]