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]
