Answer:
Complex roots for polynomials equations will always come in conjugate pairs. Since 1-2i is a root, 1+2i will also be a root. (Just switch the sign of the imaginary part to its opposite.)
Step-by-step explanation:
So far we have 1-2i and 1+2i as roots. We need two more to make a 4th degree polynomial.
x=1 with multiplicity of 2 means that we count the root twice when we write down the polynomial: (x - 1)(x - 1) = (x - 1)2
Putting it all together, we can construct our polynomial of degree 4 as
y = f(x) = (x - 1)(x - 1)[x - (1-2i)][x - (1+2i)]
Multiply the factors (FOIL):
[x - (1-2i)][x - (1+2i)] =
x2 - (1+2i)x - (1-2i)x + (1-2i)(1+2i) =
x2 - x - 2ix - x + 2ix + (1 + 2i - 2i + 4) =
x2 - 2x + 5
(x - 1)(x - 1) = x2 - x - x + 1 = x2 - 2x + 1
Now multiply the two underlined expressions:
y = (x2 - 2x + 5)(x2 - 2x + 1)
= x4 - 2x3 + 5x2 - 2x3 + 4x2 - 10x + x2 - 2x + 5 by multiplying term by term
= x4 - 4x3 + 10x2 -12x + 5 by collecting like terms.
