To simplify the rational expression [tex]\(\frac{x^2 + 2x - 35}{x^2 + 4x - 21}\)[/tex], we need to factor both the numerator and the denominator separately and then reduce any common factors.
### Step-by-Step Solution:
1. Factor the Numerator:
The numerator is [tex]\(x^2 + 2x - 35\)[/tex].
We need to find two numbers that multiply to [tex]\(-35\)[/tex] and add up to [tex]\(2\)[/tex]. These numbers are [tex]\(7\)[/tex] and [tex]\(-5\)[/tex].
Therefore, we can factor the numerator as:
[tex]\[
x^2 + 2x - 35 = (x - 5)(x + 7)
\][/tex]
2. Factor the Denominator:
The denominator is [tex]\(x^2 + 4x - 21\)[/tex].
We need to find two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(4\)[/tex]. These numbers are [tex]\(7\)[/tex] and [tex]\(-3\)[/tex].
Therefore, we can factor the denominator as:
[tex]\[
x^2 + 4x - 21 = (x - 3)(x + 7)
\][/tex]
3. Simplify the Expression:
Now, we have:
[tex]\[
\frac{x^2 + 2x - 35}{x^2 + 4x - 21} = \frac{(x - 5)(x + 7)}{(x - 3)(x + 7)}
\][/tex]
Notice that [tex]\((x + 7)\)[/tex] is a common factor in both the numerator and the denominator. We can cancel this factor:
[tex]\[
\frac{(x - 5)(x + 7)}{(x - 3)(x + 7)} = \frac{x - 5}{x - 3} \quad \text{(for \(x \neq -7\))}
\][/tex]
Therefore, the simplified form of the given rational expression is:
[tex]\[
\frac{x - 5}{x - 3}
\][/tex]
To answer the multiple-choice question:
- Divide the numerator and denominator by [tex]\((x - 21)\)[/tex]: Not correct.
- Divide the numerator and denominator by [tex]\((x + 7)\)[/tex]: Correct. We've reduced the expression by the common factor [tex]\((x + 7)\)[/tex].
- Divide the numerator and denominator by [tex]\((x - 3)\)[/tex]: Not applicable directly.
- Divide the numerator and denominator by [tex]\((x - 5)\)[/tex]: Not applicable directly.
So, the correct operation to simplify the given expression is to divide the numerator and denominator by [tex]\((x + 7)\)[/tex].
