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18. Which of the following is one of the two binomial factors of [tex]$6s^2 + 40s - 64$[/tex]?


Sagot :

To find the binomial factors of the quadratic expression [tex]\(6s^2 + 40s - 64\)[/tex], we can follow the steps for factoring a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex].

1. Identify the coefficients:
- [tex]\(a = 6\)[/tex]
- [tex]\(b = 40\)[/tex]
- [tex]\(c = -64\)[/tex]

2. Look for two numbers that multiply to [tex]\(a \cdot c\)[/tex] (which is [tex]\(6 \cdot -64 = -384\)[/tex]) and add to [tex]\(b\)[/tex] (which is [tex]\(40\)[/tex]).

The pair of numbers that meet these criteria is [tex]\(48\)[/tex] and [tex]\(-8\)[/tex], because:
[tex]\[ 48 \cdot (-8) = -384 \quad \text{and} \quad 48 + (-8) = 40 \][/tex]

3. Rewrite the middle term using these numbers:
[tex]\[ 6s^2 + 48s - 8s - 64 \][/tex]

4. Factor by grouping:
[tex]\[ 6s(s + 8) - 8(s + 8) \][/tex]

5. Factor out the common binomial factor:
[tex]\[ (6s - 8)(s + 8) \][/tex]

6. Simplify the first binomial:
[tex]\[ (3s - 4)(s + 8) \][/tex]

Therefore, the quadratic expression [tex]\(6s^2 + 40s - 64\)[/tex] factors into [tex]\((3s - 4)(s + 8)\)[/tex].

Hence, the binomial factors are [tex]\(3s - 4\)[/tex] and [tex]\(s + 8\)[/tex].

Among the provided choices, the best answer for one of the binomial factors of [tex]\(6s^2 + 40s - 64\)[/tex] is:

[tex]\[ \boxed{3s - 4} \][/tex]
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