To determine one of the binomial factors of the polynomial [tex]\( 6s^2 + 40s - 64 \)[/tex], let's follow the factoring process step-by-step.
1. Identify the polynomial to be factored:
[tex]\[
6s^2 + 40s - 64
\][/tex]
2. Assume that the polynomial can be factored into the product of two binomials:
[tex]\[
(as + b)(cs + d)
\][/tex]
3. The expanded form of the binomials:
[tex]\[
(as + b)(cs + d) = acs^2 + (ad + bc)s + bd
\][/tex]
4. Compare the terms with the original polynomial:
- [tex]\( ac = 6 \)[/tex] (coefficient of [tex]\( s^2 \)[/tex])
- [tex]\( ad + bc = 40 \)[/tex] (coefficient of [tex]\( s \)[/tex])
- [tex]\( bd = -64 \)[/tex] (constant term)
5. Solve for [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex]:
- Through trial and error and logical reasoning, we can find such values that satisfy all these equations simultaneously.
- The correct factorization is:
[tex]\[
6s^2 + 40s - 64 = 2(s + 8)(3s - 4)
\][/tex]
6. Identify one of the binomial factors:
- From the factorization [tex]\( 2(s + 8)(3s - 4) \)[/tex], the two binomial factors are: [tex]\( (s + 8) \)[/tex] and [tex]\( (3s - 4) \)[/tex].
Thus, one of the binomial factors of [tex]\( 6s^2 + 40s - 64 \)[/tex] is [tex]\( (s + 8) \)[/tex]. Another binomial factor is [tex]\( (3s - 4) \)[/tex].
The correct answer is:
[tex]\[
(s + 8)
\][/tex]
