Form a polynomial function with the given zeros

Question

Zeros: -4,2+i

Answer 1

If a complex number (2+i) is a root of a polynomial function then so is its conjugate, so add 2-i to the list. If the number "a" is a root, then the polynomial (x-a) is a factor of the polynomial.

 

So the factored form of the target polynomial looks like this:

 

[x+4][x-(2+i)][x-(2-i)]

 

Multiplying these 3 factors together results in f(x) = x³-11x+20.

 

Perhaps multiplying the binomials containing the conjugates should be the first step (using FOIL).

 

x² -x(2-i) -x(2+i) +(2-i)(2+i) = x²-2x+ix -2x -ix +4 -i² = x² -4x +5

 

Now multiply that trinomial by (x+4) to get one of the infinitely many polynomials that has the indicated 3 roots.