Let's solve this problem in two parts as provided.
### Part (a)
We are given that when the polynomial [tex]\( f(x) = x^3 - 2x + a \)[/tex] is divided by [tex]\( x - 2 \)[/tex], the remainder is 7. According to the Remainder Theorem, the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by [tex]\( (x - c) \)[/tex] is [tex]\( f(c) \)[/tex].
Given:
[tex]\[ f(x) = x^3 - 2x + a \][/tex]
[tex]\[ f(2) = 7 \][/tex]
Substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[ f(2) = 2^3 - 2 \cdot 2 + a \][/tex]
[tex]\[ 7 = 8 - 4 + a \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ 7 = 4 + a \][/tex]
[tex]\[ a = 3 \][/tex]
So, the value of [tex]\( a \)[/tex] is 3.
### Part (b)
Using the value of [tex]\( a \)[/tex] found in part (a), we need to find the remainder when the polynomial [tex]\( g(x) = 2x^3 + x^2 + ax - 5 \)[/tex] is divided by [tex]\( x + 1 \)[/tex]. According to the Remainder Theorem, the remainder of the division of a polynomial [tex]\( g(x) \)[/tex] by [tex]\( (x + 1) \)[/tex] is [tex]\( g(-1) \)[/tex].
Given:
[tex]\[ g(x) = 2x^3 + x^2 + ax - 5 \][/tex]
Using [tex]\( a = 3 \)[/tex] from part (a), we have:
[tex]\[ g(x) = 2x^3 + x^2 + 3x - 5 \][/tex]
Substitute [tex]\( x = -1 \)[/tex] into the polynomial:
[tex]\[ g(-1) = 2(-1)^3 + (-1)^2 + 3(-1) - 5 \][/tex]
[tex]\[ g(-1) = 2(-1) + 1 + (-3) - 5 \][/tex]
[tex]\[ g(-1) = -2 + 1 - 3 - 5 \][/tex]
[tex]\[ g(-1) = -9 \][/tex]
So, the remainder when [tex]\( 2x^3 + x^2 + 3x - 5 \)[/tex] is divided by [tex]\( x + 1 \)[/tex] is -9.
### Summary
- The value of [tex]\( a \)[/tex] is [tex]\( 3 \)[/tex].
- The remainder when [tex]\( 2x^3 + x^2 + 3x - 5 \)[/tex] is divided by [tex]\( x + 1 \)[/tex] is [tex]\( -9 \)[/tex].
