form a polynomial

Question

form a polynomial with the degree of 5 and zeros -3;-i;4+i

Answer 1

Given three of the roots necessary for the polynomial of degree five, we use the Fundamental Theorem of Algebra to begin the construction.

P(x) = f(x)*(x + 3)(x + i)(x - (4+i))

Since complex roots always come in pairs, we know the final two roots we can use...

P(x) = (x + (4+i))(x - i)(x + 3)(x + i)(x - (4+i))

Simplifying...

P(x) = (x² + 17)(x² + 1)(x + 3)

P(x) = (x⁴ + 18x² + 17)(x + 3)

P(x) = x⁵ + 3x⁴ + 18x³ + 18x² + 17x + 51