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Question Number 68422 by ajfour last updated on 10/Sep/19

	Answered by ajfour last updated on 10/Sep/19

	$C = 10 m^{3} p^{2} + b (m^{3} + 6 m^{2} p + 3 {m p}^{2})$ $+ c (3 m^{2} + 6 m p + p^{2}) + d (6 m + 4 p)$ $+ 10 e$ $D = 10 m^{2} p^{3} + b (3 m^{2} p + 6 {m p}^{2} + p^{3})$ $+ c (m^{2} + 6 m p + 3 p^{2}) + d (4 m + 6 p)$ $+ 10 e$ $\underline{C - D = 0 g i v e s}$ $10 m^{2} p^{2} (m - p)$ $+ b [(m^{3} - p^{3}) + 6 m p (m - p) - 3 m p (m - p)]$ $+ c [3 (m^{2} - p^{2}) - (m^{2} - p^{2})]$ $+ d [6 (m - p) - 4 (m - p)] + 0 = 0$ $I f m w o u l d b e p t h e n z = m = p$ $o t h e r w i s e d i v i d i n g b y m - p g i v e s$ $10 m^{2} p^{2} + b (m^{2} + 4 m p + p^{2})$ $+ c [2 (m + p)] + 2 d = 0 . . . . . (I)$ $(i f m \neq - p e v e n)$ $e q . (i) \times (m + p) g i v e s$ $10 m^{2} p^{2} (m + p)$ $+ b [m^{3} + 4 m^{2} p + {m p}^{2} + m^{2} p + 4 {m p}^{2}$ $+ p^{3}] + c [2 m^{2} + 4 m p + 2 p^{2}]$ $+ d (2 m + 2 p) = 0 . . . . . . (i)$ $\underline{N o w C + D = 0 g i v e s}$ $10 m^{2} p^{2} (m + p) +$ $b [m^{3} + p^{3} + 6 m p (m + p) + 3 m p (m + p)]$ $+ c (4 m^{2} + 4 p^{2} + 12 m p)$ $+ d (10 m + 10 p) + 20 e = 0 . . . . (I I)$ $\underline{(I I) - (i) g i v e s}$ $b [4 m p (m + p)] +$ $c [2 m^{2} + 8 m p + 2 p^{2}] +$ $d [8 (m + p)] + 20 e = 0$ $\Rightarrow$ $2 b m p (m + p) + c [{(m + p)}^{2} + 2 m p]$ $+ 4 d (m + p) + 10 e = 0 . . . . (I I I)$ $N o w l e t s t r a n s f o r m e q . (I) & (I I I)$ $l e t m + p = s, a n d m p = g$ $\underline{E q . (I)}$ $10 g^{2} + b (s^{2} + 2 g) + 2 c s + 2 d = 0$ $\underline{E q . (I I I)}$ $2 b g s + c (s^{2} + 2 g) + 4 d s + 10 e = 0$ $\Rightarrow g = - \frac{{c s}^{2} + 4 d s + 10 e}{2 (b s + c)}$ $\underline{s u b s t i t u t i n g f o r g i n e q . (I)}$ $10 {({c s}^{2} + 4 d s + 10 e)}^{2} -$ $4 b ({c s}^{2} + 4 d s + 10 e) (b s + c)$ $+ 4 ({b s}^{2} + 2 c s + 2 d) {(b s + c)}^{2} = 0$ $\Rightarrow$ $5 {({c s}^{2} + 4 d s + 10 e)}^{2}$ $- 2 b (b s + c) ({c s}^{2} + 4 d s + 10 e)$ $+ 2 {(b s + c)}^{2} ({b s}^{2} + 2 c s + 2 d) = 0$ $T h i s i s a d e g r e e 4 e q . i n s .$ $S o i t s e e m s i s h a l l o b t a i n s$ $a n d g = - \frac{{c s}^{2} + 4 d s + 10 e}{2 (b s + c)} .$ $m, p = \frac{s}{2} \pm \sqrt{\frac{s^{2}}{4} - g}$ $N o w w e t h e n h a v e$ ${A t}^{5} + {B t}^{4} + E t + F = 0$ $o r t^{5} + (\frac{B}{A}) t^{4} + (\frac{E}{A}) t + \frac{F}{A} = 0$ $b u t i f m = p c l e a r l y z = m$ $h e n c e a l l c o e f f i c i e n t s g e t z e r o$ $a n d w e n e e d t o c h a n g e$ $t o z = \frac{m t + p}{t - 1}; t h i s c a s e i s h a l l t a k e$ $u p i n a n o t h e r p o s t . .$
	Commented by TawaTawa last updated on 10/Sep/19

	$wow, God bless you sir and increase your knowledge . I will learn$ $this with an example .$
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