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Question Number 208836 by Adeyemi889 last updated on 24/Jun/24

	Answered by Rasheed.Sindhi last updated on 25/Jun/24

	${A x}^{3} - A + {B x}^{2} + B x + B + (C x + D) (x^{2} - 2 x + 1)$ $= {A x}^{3} - A + {B x}^{2} + B x + B + {C x}^{3} + {D x}^{2} - 2 {C x}^{2} - 2 D x + C x + D$ $= (A + C) x^{3} + (B + D - 2 C) x^{2} + (B - 2 D + C) x + (- A + B + D)$ $= x^{3} + 5 x^{2} + 4 x + 5$ $C o m p a r i n g c o e f f i c i e n t s :$ $A + C = 1 . . . i$ $B + D - 2 C = 5 . . . i i$ $B + C - 2 D = 4 . . . i i i$ $- A + B + D = 5 . . . i v$ $i v \Rightarrow B + D = A + 5$ $i i \Rightarrow A + 5 - 2 C = 5 \Rightarrow A = 2 C$ $i \Rightarrow 3 C = 1 \Rightarrow C = 1 / 3 \Rightarrow A = 2 / 3$ $i i i \Rightarrow B + C - 2 D = 4$ $\Rightarrow B + 1 / 3 - 2 D = 4$ $\Rightarrow B - 2 D = 4 - 1 / 3 = 11 / 3$ $B + D = 2 / 3 + 5 = 17 / 3$ $- 3 D = - 6 / 3 \Rightarrow D = 2 / 3$ $B + D = 17 / 3 \Rightarrow B = 17 / 3 - 2 / 3 = 5$ $\Rightarrow B = 5$
	Commented by Adeyemi889 last updated on 25/Jun/24

	$y e s s i r t h e l a s t p a r t o f t h e q u e s t i o n (C x + B) i s w i t h {(X - 1)}^{2}$ $i n o t i c e i n u r s o l u t i o n s i r u o n l y w r o t e (C x + B)$
	Commented by Rasheed.Sindhi last updated on 25/Jun/24

	$S o r r y s i r, I {d i d n}^{'} t n o t i c e {(x - 1)}^{2} w h i c h$ $i s o n n e x t l i n e . I^{'} l l g o t o c o r r e c t i t .$ $C o r r e c t e d$
	Answered by Rasheed.Sindhi last updated on 25/Jun/24

	$A n o t h e r w a y$ $x^{3} + 5 x^{2} + 4 x + 5$ $= A (x^{3} - 1) + B (x^{2} + x + 1) + (C x + D) {(x - 1)}^{2}$ $T h e a b o v e i s a n i d e n t i t y s o {i t}^{'} s$ $c o r r e c t f o r a l l v a l u e s o f x .$ $x = 0 :$ $5 = - A + B + D . . . . (i)$ $x = 1 :$ $15 = 3 B \Rightarrow B = 5$ $x = 2 :$ $41 = 7 A + 7 B + 2 C + D . . . . (i i)$ $x = - 1 :$ $5 = - 2 A + B - 4 C + 4 D . . . . (i i i)$ $(i) \Rightarrow - A + D = 0 \Rightarrow A = D . . . (i v)$ $(i i) \Rightarrow 7 A + 2 C + D = 6$ $\Rightarrow 8 A + 2 C = 6 . . . (v)$ $(i i i) \Rightarrow - 2 A - 4 C + 4 D = 0$ $\Rightarrow 2 A - 4 C = 0$ $\Rightarrow A = 2 C$ $(v) \Rightarrow 8 A + A = 6 \Rightarrow A = \frac{2}{3}$ $(i v) \Rightarrow D = \frac{2}{3}$ $\Rightarrow C = \frac{1}{3}$
	Commented by Adeyemi889 last updated on 26/Jun/24

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