Find the polynomial function f with real coefficients that has the given degree, zeros, and solution

Question

Degree: 4. Zeros: -3, 1, i. Solution point: f(0) = -6.

Answer 1

Since i is a root so is -i.

Since -3 is a root (x+3) is a factor.

Since 1 is a root (x - 1 ) is a factor.

Since i is a root (x - i) is a factor.

Since -i is a root (x + i) is a factor.

So f(x) can be written as:

f(x) = A (x+3)(x-1)(x - i) (x + i)

f(x) = A (x²+2x -3) (x²+1)

= A (x⁴+2x³-2x²+2x-3)

Since f(0) = -6, plug 0 in for x and set f(x) = -6.

-6=A(-3)

A=2

Go ahead and distribute the 2 to get:

f(x) = 2x⁴+4x³-4x²+4x-6

Visit the following Desmos graph for confirmation:

desmos.com/calculator/makqrdljsx

Note:

(x - i)(x + i) results in a difference of squares: x²-i², and since i²=-1 this is the same as x²+1