Let's go through each part step-by-step.
Part 1: Evaluate [tex]\( g(2) \)[/tex]
Given the function:
[tex]\[ g(x) = 6x^4 - 2x^3 + 3x^2 - 2 \][/tex]
To evaluate [tex]\( g(2) \)[/tex], we need to substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ g(2) = 6(2)^4 - 2(2)^3 + 3(2)^2 - 2 \][/tex]
[tex]\[ g(2) = 6(16) - 2(8) + 3(4) - 2 \][/tex]
[tex]\[ g(2) = 96 - 16 + 12 - 2 \][/tex]
[tex]\[ g(2) = 90 \][/tex]
So,
[tex]\[ g(2) = 90 \][/tex]
Part 2: Determine the remainder when [tex]\( g(x) \)[/tex] is divided by [tex]\( (x-2) \)[/tex]
To find the remainder when [tex]\( g(x) \)[/tex] is divided by [tex]\( (x-2) \)[/tex], we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial [tex]\( g(x) \)[/tex] by [tex]\( (x-a) \)[/tex] is [tex]\( g(a) \)[/tex].
In this case, for [tex]\( (x-2) \)[/tex], [tex]\( a = 2 \)[/tex]. Therefore, the remainder when [tex]\( g(x) \)[/tex] is divided by [tex]\( (x-2) \)[/tex] is [tex]\( g(2) \)[/tex].
We have already calculated [tex]\( g(2) \)[/tex] in Part 1:
[tex]\[ g(2) = 90 \][/tex]
So, the remainder when [tex]\( g(x) \)[/tex] is divided by [tex]\( (x-2) \)[/tex] is:
[tex]\[ \text{The remainder is } 90 \][/tex]
To summarize:
Part 1:
[tex]\[ g(2) = 90 \][/tex]
Part 2:
[tex]\[ \text{The remainder is } 90 \][/tex]
