To determine which binomial is a factor of the trinomial [tex]\( x^2 + 2x - 35 \)[/tex], we proceed with factoring the trinomial. Here is the detailed step-by-step solution:
1. Recognize the form of the trinomial: Given [tex]\( x^2 + 2x - 35 \)[/tex], we look for two binomials whose product results in this trinomial.
2. Consider the general factorization method for trinomials of the form [tex]\( x^2 + bx + c \)[/tex]: We need two numbers that multiply to [tex]\(-35\)[/tex] (the constant term) and add to [tex]\(2\)[/tex] (the coefficient of the linear term [tex]\(x\)[/tex]).
3. Find the pair of numbers: Start listing the factor pairs of [tex]\(-35\)[/tex]:
- [tex]\((1, -35)\)[/tex]
- [tex]\((-1, 35)\)[/tex]
- [tex]\((5, -7)\)[/tex]
- [tex]\((-5, 7)\)[/tex]
4. Check the sum of each pair: We are looking for a pair that sums to [tex]\(2\)[/tex]:
- [tex]\(1 + (-35) = -34\)[/tex]
- [tex]\(-1 + 35 = 34\)[/tex]
- [tex]\(5 + (-7) = -2\)[/tex]
- [tex]\(-5 + 7 = 2\)[/tex]
The pair [tex]\((-5, 7)\)[/tex] yields a sum of [tex]\(2\)[/tex], which matches the coefficient of [tex]\(x\)[/tex].
5. Write the binomials: The trinomial then factors to:
[tex]\[
x^2 + 2x - 35 = (x - 5)(x + 7)
\][/tex]
6. Identify the correct binomial: From the factorization, we can see that the factors of the trinomial [tex]\( x^2 + 2x - 35 \)[/tex] are [tex]\( (x - 5) \)[/tex] and [tex]\( (x + 7) \)[/tex].
7. Match with given options: The options provided are:
- A. [tex]\( x + 1 \)[/tex]
- B. [tex]\( x - 7 \)[/tex]
- C. [tex]\( x - 1 \)[/tex]
- D. [tex]\( x + 7 \)[/tex]
We observe that the binomial [tex]\( x + 7 \)[/tex] is a factor of the trinomial.
Thus, the correct answer is:
[tex]\[
\boxed{x + 7}
\][/tex]
