Remainder Theorem Question?

The remainder when divided by (x - 1) is the same as the value of the function when x = 1.

Since x = 1 here, all of the x-terms pretty much knock themselves out and you are left with the sum of the first 50 terms of an arithmetic sequence of:

a(n) = 2n

Knowing that we can then use the sum of the first n terms of the sequence to find the answer:

S(n) = [ a(1) + a(n)] n / 2

We know the first term is 2 and the last term is 100 and there are 50 terms, so:

S(50) = [ a(1) + a(50)] * 50 / 2
S(50) = (2 + 100) * 25
S(50) = 102 * 25
S(50) = 2550

P(x) = 2x + 4x² + 6x³ + ... + 100x⁵⁰

When P(x) is divided by (x - 1), remainder
= P(1)
= 2(1) + 4(1)² + 6(1)³ + ... + 100(1)⁵⁰
= 2 + 4 + 6 + … + 100
= 2 × (1 + 2 + 3 + … + 50)
= 2 × (1 + 50) × 50 / 2
= 2550