Here we use the remainder theorem. As ax^3 + bx^2 + cx + d is divided by (x-2).
When f(x) is divided by (x-a) the remainder is given by f(a).
So here we have the remainder as f(x) = ax^3 + bx^2 + cx + d for x = 2.
=> a*2^3 + b*2^2 + c*2 + d
=> a* 8 + b*4 + c*2 + d
=> 8a + 4b + 2c +d
Therefore the remainder when ax^3 + bx^2 + cx + d is divided by (x-2) is 8a + 4b + 2c +d.
Since the polynomial is divided by the binomial x-2, the reminder is a constant.
We'll write the division with reminder:
ax^3+bx^2+cx+d = (x-2)(ex^2 + fx + g) + h
The reminder R(x) = h
The fundamental theorem of algebra states that the reminder of a polynomial divided by a binomial x-a is:
P(a) = R(a)
We'll substitute x by 2 in the expression of polynomial:
P(2) = a*2^3+b*2^2+c*2+d
P(2) = 8a + 4b + 2c + d
The reminder is:
R(2) = 8a + 4b + 2c + d