To find the value of [tex]\( a \)[/tex] such that 2 is a zero of the polynomial [tex]\( f(x) = a x^2 - 3(a-1) x - 1 \)[/tex], follow these steps:
1. Substitute [tex]\( x = 2 \)[/tex] into the polynomial: Since 2 is a zero of the polynomial, it should satisfy the equation [tex]\( f(2) = 0 \)[/tex].
[tex]\[
f(2) = a (2)^2 - 3(a-1) (2) - 1
\][/tex]
2. Simplify the expression: Substitute [tex]\( x = 2 \)[/tex] into the polynomial and simplify.
[tex]\[
f(2) = a (2^2) - 3(a-1) (2) - 1 = a (4) - 3(a-1) (2) - 1
\][/tex]
[tex]\[
f(2) = 4a - 3 [2(a - 1)] - 1 = 4a - 3 (2a - 2) - 1
\][/tex]
[tex]\[
f(2) = 4a - 6a + 6 - 1 = -2a + 5
\][/tex]
3. Set the simplified expression equal to 0: Since 2 is a zero of the polynomial, we have:
[tex]\[
-2a + 5 = 0
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
[tex]\[
-2a + 5 = 0
\][/tex]
[tex]\[
-2a = -5
\][/tex]
[tex]\[
a = \frac{5}{2}
\][/tex]
Hence, the value of [tex]\( a \)[/tex] is [tex]\(\boxed{\frac{5}{2}}\)[/tex].
