Given-3-2i-and-1-i-are-the-two-of-roots-of-the-equation-ax-4-bx-3-cx-3-dx-e-find-the-values-of-a-b-c-d-and-e-

Question Number 50970 by peter frank last updated on 22/Dec/18

G i v e n 3 - 2 i a n d 1 + i

a r e t h e t w o o f r o o t s o f

t h e e q u a t i o n

{a x}^{4} + {b x}^{3} + {c x}^{3} + d x + e

f i n d t h e v a l u e s o f

a, b, c, d a n d e

Answered by peter frank last updated on 23/Dec/18

l e t z_{1} = 3 - 2 i z_{2} = 1 + i

{\bar{z}}_{1} = 3 + 2 i {\bar{z}}_{2} = 1 - i

x^{2} - (3 + 2 i + 3 - 2 i) x + (3 + 2 i) (3 - 2 i) = 0

x^{2} - 6 x + 13 = 0 \dots \dots (i)

t h e s a m e w a y

x^{2} - (1 + i + 1 - i) x + (1 + i) (1 - i) = 0

x^{2} - 2 x + 2 = 0

m u l t i p l y (i) \times (i i)

(x^{2} - 6 x + 13) (x^{2} - 2 x + 2) = 0

x^{4} - 8 x^{3} + 27 x^{2} - 38 x + 26 = 0 \dots . . (i i i)

↓

{a x}^{4} + {b x}^{3} + {c x}^{2} + d x + 26

a = 1 b = - 8 c = 27 d = - 38

e = 26