Expand (1+x)^3. Express (1+i)^3 in the form a+bi. Hence, or otherwise, verify that x = 1+i satisfies the equation: x^3+2*x-4i = 0.

First, we factor out one (x+1). (1+x)^3 = (x+1)^2(x+1)= Then, we expand using the formula (a+b)^2 = a^2 + 2ab +b^2: =(x^2+2x+1)(x+1)= Then we multiply: = x^3 + 2x^2 + x + x^2 + 2x + 1 We sum the terms with the same power and we get: (1+x)^3 = x^3 + 3x^2 + 3x +1. For the next part, we expand using the formula we just calculated: (1+i)^3 = i^3 + 3i^2 + 3i + 1 = We replace i^2 with -1: = -i -3 +3i +1 Next, we bring this to the a+bi form: (1+i)^3 = 2i-2. To check if x = i+1 is a root of that equation, we replace x with i+1 and see if the result is 0: (1+i)^3 + 2(1+i) - 4i = We use the form of (1+i)^3 which we calculated above and expand the bracket: 2i-2 + 2i+2 -4i = 0.