How do you write a polynomial with zeros 8, -i, i and leading coefficient 1?

1 Answer
Mar 2, 2017

color(green)(x^3-8x^2+x-8)x38x2+x8

Explanation:

If the polynomial has zeros 8, -i, and i8,i,andi
then it has factors cxx(x-8)(x+i)(x-i)c×(x8)(x+i)(xi) where cc is a constant.

Since we are asked for a polynomial with a leading coefficient of 11,
c=1c=1 and we can ignore it from here on.

color(blue)((x+i)(x-i))=x^2-i^2=x^2-(-1)=color(blue)(x^2+1)(x+i)(xi)=x2i2=x2(1)=x2+1
and
(x-8)color(blue)((x+i)(x-i))=(x-8)xxcolor(blue)(""(x^2+1))=x^3-8x^2+x-8(x8)(x+i)(xi)=(x8)×(x2+1)=x38x2+x8