Answer and Step-by-step explanation:
polynomial P with the lowest possible degree
solutions: a)−2, 3,−4i,4i b)−1, 3i c)0,2,1 +i,1-i
The polynomial with the lowest degree that has the given solutions is a multiplication of the monomials (x-solution). And with complex roots, we always have a number and its conjugate
a)−2, 3,−4i,4i
P(x) = (x-(-2))*(x-3)*(x-(-4i))*(x-4i) = (x+2)(x-3)(x-4i)(x+4i) = (x²-3x+2x-6)(x²-16i²) = (x²-x-6)(x²+16) = x⁴+16x²-x³-16x-6x²-96 = x⁴-x³+10x²-16x-96
b)−1, 3i
(x-(-1))*(x-3i)*(x-(-3i)) = (x+1)(x-3i)(x+3i) = (x+1)(x²-9i²) = (x+1)(x²+9) = x³+9x+x²+9
c) 0,2,1 +i,1-i
(x-0)(x-2)(x-(1+i))(x-(1-i)) = x(x-2)(x-1-i)(x-1+i) = (x²-2x)(x²-x+xi-x+1-i-xi+i-i²) =
(x²-2x)(x²-x-x+1+1) = (x²-2x)(x²-2x+2) = x⁴-2x³+2x²-2x³+4x²-4x = x⁴-4x³+6x²-4x
