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Which of the binomials below is a factor of this trinomial?

[tex]\[ x^2 + 2x - 35 \][/tex]

A. [tex]\( x + 1 \)[/tex]
B. [tex]\( x - 7 \)[/tex]
C. [tex]\( x - 1 \)[/tex]
D. [tex]\( x + 7 \)[/tex]


Sagot :

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]