To find the value of [tex]\( k \)[/tex] when [tex]\((x-1)\)[/tex] is a factor of the polynomial [tex]\( p(x) = kx^2 - 3x + k \)[/tex], we can use the Factor Theorem. The Factor Theorem states that if [tex]\((x-a)\)[/tex] is a factor of [tex]\( p(x) \)[/tex], then [tex]\( p(a) = 0 \)[/tex].
In this case, since [tex]\((x-1)\)[/tex] is a factor, we set [tex]\( p(1) = 0 \)[/tex].
Let's evaluate [tex]\( p(1) \)[/tex]:
[tex]\[ p(1) = k(1)^2 - 3(1) + k \][/tex]
Simplifying, we get:
[tex]\[ p(1) = k - 3 + k \][/tex]
Combining like terms:
[tex]\[ p(1) = 2k - 3 \][/tex]
According to the Factor Theorem, this must equal 0 since [tex]\((x-1)\)[/tex] is a factor:
[tex]\[ 2k - 3 = 0 \][/tex]
Now, solve for [tex]\( k \)[/tex]:
[tex]\[ 2k = 3 \][/tex]
[tex]\[ k = \frac{3}{2} \][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{\frac{3}{2}} \)[/tex].
