Question

b) Show that (1+i) is a zero of F(x) = 2x^5 - 9x^4 +12x^3 -4x^2 - 8x +4 c) Find all of the ZEROES of F(x)

Answer 1

b) Give F(x) = 2005 _ 9 oc" + 12x²_40c²-83ct4 - Now Hi is a sero of the fox) if on substituting = Hi in f(x) it gives f (ti)=0 (1+i)? = 12+12+2i = 1+2i = 21 (1+i)} = (+) (1+i)= 2i (iti)= -2 +21 (1+i)* = 1+7)+ (1+i) = 2:)(23) = -4 (1+i)= (1+)" (i+i) = -4(1+i) = -4-li Plug it in @ in the place of a F (hti) = 2(4-42) -9(-4) +121-2+2:) - 42: )-tv)+4 --8-81+36 -24+241-81-8-80th F (Hi) =0 Thus X= ti is a sers of flood.

(© . Since Complex Heros are occurs in Conjugate bair therefore x=1-1 is also the seas of food. Now X= Hii and xal-i 26-1-1=0 26-1 tiro & (6-1-1) (21-1+i)=0 = 6c-1/²-i² so ... = 22222 +1 +170 I let g (%) = 2c2_2x+2=0 Divide FG) by g(x) it reduces the polynomial into cubic. 2005-904+12x3_42?_84+4,233-582_2X+2 2c²- 221+ 2

Now find other 3 years using by solving for 226?- Sz 2-2x+2=0 = By guess we found a = 1/2 is the solution of the above equation 26=-2x-170:-0 Diride ® by 6 it give a quadratic equation - 00² -22-2=0 quadratic formula x=-62) I JE2²_4X1XE21 - 1 3 Thus, all the serps of f(x) are ! - Hi, 1-1, 1 ts and 1-3 - 281