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


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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

	$l e t z_{1} = 3 - 2 i z_{2} = 1 + i$ ${\bar{z}}_{1} = 3 + 2 i {\bar{z}}_{2} = 1 - i$ $x^{2} - (3 + 2 i + 3 - 2 i) x + (3 + 2 i) (3 - 2 i) = 0$ $x^{2} - 6 x + 13 = 0 . . . . . . (i)$ $t h e s a m e w a y$ $x^{2} - (1 + i + 1 - i) x + (1 + i) (1 - i) = 0$ $x^{2} - 2 x + 2 = 0$ $m u l t i p l y (i) \times (i i)$ $(x^{2} - 6 x + 13) (x^{2} - 2 x + 2) = 0$ $x^{4} - 8 x^{3} + 27 x^{2} - 38 x + 26 = 0 . . . . . (i i i)$ $↓$ ${a x}^{4} + {b x}^{3} + {c x}^{2} + d x + 26$ $a = 1 b = - 8 c = 27 d = - 38$ $e = 26$
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