To determine which of the given expressions is equal to the rational expression [tex]\(\frac{x^2 - 49}{x - 7}\)[/tex] when [tex]\(x \neq 7\)[/tex], follow these steps for simplification and comparison:
1. Express the numerator in factored form:
The numerator [tex]\(x^2 - 49\)[/tex] is a difference of squares. Recall the difference of squares formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
In this case, [tex]\(a = x\)[/tex] and [tex]\(b = 7\)[/tex]. Thus,
[tex]\[
x^2 - 49 = (x - 7)(x + 7)
\][/tex]
2. Substitute the factored form into the original expression:
[tex]\[
\frac{x^2 - 49}{x - 7} = \frac{(x - 7)(x + 7)}{x - 7}
\][/tex]
3. Simplify the expression by canceling the common factor:
Provided that [tex]\(x \neq 7\)[/tex],
[tex]\[
\frac{(x - 7)(x + 7)}{x - 7} = x + 7
\][/tex]
Therefore, the rational expression [tex]\(\frac{x^2 - 49}{x - 7}\)[/tex] simplifies to [tex]\(x + 7\)[/tex] when [tex]\(x \neq 7\)[/tex].
Now, let's compare this simplified form to the given options:
- A. [tex]\(\frac{x+7}{x-7}\)[/tex]
- B. [tex]\(x+7\)[/tex]
- C. [tex]\(\frac{1}{x+7}\)[/tex]
- D. [tex]\(x-7\)[/tex]
Clearly, the expression [tex]\(x + 7\)[/tex] matches option B.
Hence, the correct answer is:
[tex]\[
\boxed{B}
\][/tex]
