a, b and c are solutions of x^3 −4x^2 −5x+8=0. Without determinating a, b and c ; calculate a+b+c.


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Question Number 121918 by greg_ed last updated on 12/Nov/20

$a, b a n d c a r e s o l u t i o n s o f x^{3} - 4 x^{2} - 5 x + 8 = 0 .$ $W i t h o u t d e t e r m i n a t i n g a, b a n d c;$ $c a l c u l a t e a + b + c .$
Commented by mr W last updated on 12/Nov/20

$a + b + c = 4$ $a b + b c + c a = - 5$ $a b c = - 8$
Commented by greg_ed last updated on 12/Nov/20

$please, mr W : {it}^{'} s not very very clear for$ $me!$
Commented by physicstutes last updated on 12/Nov/20

$Generally if α, β and γ are roots of the equation,$ ${a x}^{3} + {b x}^{2} + c x + d = 0$ $then α + β + γ = - \frac{b}{a}$ $α β γ = - \frac{d}{a}$
Commented by greg_ed last updated on 12/Nov/20

$thanks, physicstutes!$
Commented by MJS_new last updated on 12/Nov/20

$(x - a) (x - b) (x - c) = 0$ $x^{3} - (a + b + c) x^{2} + (a b + a c + b c) x - a b c = 0$ $x^{3} - 4 x^{2} + (- 5) x - (- 8) = 0$
Commented by greg_ed last updated on 12/Nov/20

$ok, MJS_new! {it}^{'} s cool!$
	Answered by Dwaipayan Shikari last updated on 12/Nov/20

	$f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + . . . + a_{n} = 0$ $I n t h i s p o l y n o m i a l s u m o f t h e r o o t s i s (- \frac{a_{1}}{a_{0}})$ $h e r e p o l y n o m i a l i s x^{3} - 4 x^{2} - 5 x + 8 = 0$ $s u m o f t h e r o o t s = - (- \frac{4}{1}) = 4$
	Commented by greg_ed last updated on 12/Nov/20

	$thanks, Dwaipayan Shikari!$
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