To determine which of the given binomials is a factor of the trinomial [tex]\( x^2 - 5x - 36 \)[/tex], we will check each binomial one by one.
### Step-by-step Solution:
1. Binomial [tex]\( x - 4 \)[/tex]:
- To check if [tex]\( x - 4 \)[/tex] is a factor of [tex]\( x^2 - 5x - 36 \)[/tex], we perform polynomial division:
- Divide [tex]\( x^2 - 5x - 36 \)[/tex] by [tex]\( x - 4 \)[/tex].
- After performing the division, if the remainder is 0, then [tex]\( x - 4 \)[/tex] is a factor. If not, it is not a factor.
2. Binomial [tex]\( x + 4 \)[/tex]:
- To check if [tex]\( x + 4 \)[/tex] is a factor of [tex]\( x^2 - 5x - 36 \)[/tex], we again perform polynomial division:
- Divide [tex]\( x^2 - 5x - 36 \)[/tex] by [tex]\( x + 4 \)[/tex].
- After performing the division, if the remainder is 0, then [tex]\( x + 4 \)[/tex] is a factor. If not, it is not a factor.
3. Binomial [tex]\( x^2 + 7 \)[/tex]:
- To check if [tex]\( x^2 + 7 \)[/tex] is a factor of [tex]\( x^2 - 5x - 36 \)[/tex], we again perform polynomial division:
- Divide [tex]\( x^2 - 5x - 36 \)[/tex] by [tex]\( x^2 + 7 \)[/tex].
- After performing the division, if the remainder is 0, then [tex]\( x^2 + 7 \)[/tex] is a factor. If not, it is not a factor.
4. Binomial [tex]\( x - 7 \)[/tex]:
- To check if [tex]\( x - 7 \)[/tex] is a factor of [tex]\( x^2 - 5x - 36 \)[/tex], we again perform polynomial division:
- Divide [tex]\( x^2 - 5x - 36 \)[/tex] by [tex]\( x - 7 \)[/tex].
- After performing the division, if the remainder is 0, then [tex]\( x - 7 \)[/tex] is a factor. If not, it is not a factor.
Upon careful examination of each of these binomials:
- [tex]\( x - 4 \)[/tex] is not a factor since the remainder from the division is not 0.
- [tex]\( x + 4 \)[/tex] is a factor because the remainder from the division is 0.
- [tex]\( x^2 + 7 \)[/tex] is not a factor since the remainder from the division is not 0.
- [tex]\( x - 7 \)[/tex] is not a factor since the remainder from the division is not 0.
Thus, the correct answer is:
B. [tex]\( x + 4 \)[/tex]
