To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the polynomial [tex]\(2x^3 - ax^2 + bx + 12\)[/tex], given that it is divisible by [tex]\((x + 1)\)[/tex] and [tex]\((x - 4)\)[/tex], we can follow these steps:
1. Substitute [tex]\(x = -1\)[/tex]:
Since [tex]\((x + 1)\)[/tex] is a factor of the polynomial, substituting [tex]\(x = -1\)[/tex] should make the polynomial equal to zero.
[tex]\[
2(-1)^3 - a(-1)^2 + b(-1) + 12 = 0
\][/tex]
Simplify the equation:
[tex]\[
-2 - a - b + 12 = 0
\][/tex]
Combine like terms:
[tex]\[
10 - a - b = 0
\][/tex]
Rearrange to isolate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a + b = 10
\][/tex]
2. Substitute [tex]\(x = 4\)[/tex]:
Similarly, since [tex]\((x - 4)\)[/tex] is a factor of the polynomial, substituting [tex]\(x = 4\)[/tex] should also make the polynomial equal to zero.
[tex]\[
2(4)^3 - a(4)^2 + b(4) + 12 = 0
\][/tex]
Simplify the equation:
[tex]\[
2 \cdot 64 - 16a + 4b + 12 = 0
\][/tex]
[tex]\[
128 - 16a + 4b + 12 = 0
\][/tex]
Combine like terms:
[tex]\[
140 - 16a + 4b = 0
\][/tex]
Rearrange to isolate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
16a - 4b = 140
\][/tex]
Simplify by dividing the entire equation by 4:
[tex]\[
4a - b = 35
\][/tex]
3. Solve the system of linear equations:
Now we have two equations:
[tex]\[
\begin{cases}
a + b = 10 \\
4a - b = 35
\end{cases}
\][/tex]
Add the two equations to eliminate [tex]\(b\)[/tex]:
[tex]\[
(a + b) + (4a - b) = 10 + 35
\][/tex]
[tex]\[
5a = 45
\][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[
a = 9
\][/tex]
Substitute [tex]\(a = 9\)[/tex] back into the first equation:
[tex]\[
9 + b = 10
\][/tex]
Solve for [tex]\(b\)[/tex]:
[tex]\[
b = 1
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are [tex]\(9\)[/tex] and [tex]\(1\)[/tex] respectively. The correct answer is:
[tex]\[ \boxed{9, 1} \][/tex]
