Find the equation

Question

gg is a degree 5 polynomial that passes through the origin, has zeros ii and 8−i8-i, and g(−3)=−14640g(-3)=-14640. Find the equation for g.

Answer 1

i is a zero means

x=i

square both sides to get

x^2=-1

x^2+1=0

x^2+1 is one factor

8-i is a factor means

x=8-i

x-8=-i

x^2-16x+64=-1

x^2-16x+65=0

x^2-16x+65 is a 2nd factor

through the origin means x is a zero

and (x-0) or x is the 3rd factor

g(x)=a(x)(x^2+1)(x^2-16x+65)

to calculate a, the coefficient of the 5th degree term, plug in x=-3

g(-3)=a(-3)(10)(9+48+65)=-3660a=-14640

a=1464/366=4

g(x)= 4x(x^2+1)(x^2-16x+65)

g(x) =(4x^3 + 4x)(x^2-16x+65)

g(x) = 4x^5 -64x^4 264x^3 -64x^2 +260x

is a 5th degree polynomial through the origin

with zeros i and 8-i and where g(-3)= -14640