Find roots of f(x)=x^4 −10x^3 +35x^2 −50x+24


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Question Number 39783 by ajfour last updated on 10/Jul/18

$F i n d r o o t s o f$ $f (x) = x^{4} - 10 x^{3} + 35 x^{2} - 50 x + 24$
	Answered by ajfour last updated on 10/Jul/18

	$l e t f (x) = {(x^{2} - 5 x + A)}^{2} - {(B x + C)}^{2}$ $= (x^{2} - 5 x + A - B x - C) (x^{2} - 5 x + B x + C)$ $i f w e s e t x = 0$ $\Rightarrow A^{2} - C^{2} = 24 = 5^{2} - 1^{2}$ $l e t A = 5; C = 1$ $i f w e s e t x = 5 t h e n$ $(A - 5 B - C) (A + 5 B + C) =$ $625 - 1250 + 875 - 250 + 24 = 24$ $\Rightarrow A^{2} - {(5 B + C)}^{2} = 24$ $o r {(5 B + 1)}^{2} = 1$ $\Rightarrow B = 0$ $h e n c e$ $f (x) = (x^{2} - 5 x + 4) (x^{2} - 5 x + 6)$ $= (x - 1) (x - 4) (x - 2) (x - 3) .$
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