If 2x^2 +11x+5 is a factor of ax^3 −17x^2 +bx−15 , then what is a and b ?


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Question Number 157416 by cortano last updated on 23/Oct/21

$I f 2 x^{2} + 11 x + 5 i s a f a c t o r o f$ ${a x}^{3} - 17 x^{2} + b x - 15, t h e n w h a t$ $i s a a n d b ?$
Commented by Rasheed.Sindhi last updated on 23/Oct/21

${a x}^{3} - 17 x^{2} + b x - 15 h a s a f a c t o r$ $\underline{2 x^{2} + 11 x + 5; a = ?, b = ?}$ $O t h e r f a c t o r = \frac{a}{2} x + \frac{- 15}{5} = \frac{a}{2} x - 3$ $▸ (2 x^{2} + 11 x + 5) (\frac{a}{2} x - 3)$ $= {a x}^{3} - 17 x^{2} + b x - 15$ $▸ {a x}^{3} + \frac{11 a}{2} x^{2} + \frac{5 a}{2} x - 6 x^{2} - 33 x - 15$ $= {a x}^{3} - 17 x^{2} + b x - 15$ $▸ {a x}^{3} + (\frac{11 a}{2} - 6) x^{2} + (\frac{5 a}{2} - 33) x - 15$ $= {a x}^{3} - 17 x^{2} + b x - 15$ $C o m p a r i n g c o e f f i c i e n t s :$ $\frac{11 a}{2} - 6 = - 17 \Rightarrow a = \frac{- 22}{11} = - 2$ $\frac{5 a}{2} - 33 = b \Rightarrow b = \frac{5 (- 2) - 66}{2} = - 38$
Commented by Tawa11 last updated on 23/Oct/21

$Weldone sir$
Commented by Rasheed.Sindhi last updated on 23/Oct/21

$T han k s miss!$ $I like your encouraging!$
	Answered by Rasheed.Sindhi last updated on 23/Oct/21

	$2 x^{2} + 11 x + 5 is factor of$ $\underline{{a x}^{3} - 17 x^{2} + b x - 15; a = ?, b = ?}$ $L e t t h e o t h e r f a c t o r i s c x + d$ $▸ (2 x^{2} + 11 x + 5) (c x + d)$ $= {a x}^{3} - 17 x^{2} + b x - 15$ $▸ 2 {c x}^{3} + (11 c + 2 d) x^{2} + (5 c + 11 d) x + 5 d$ $= {a x}^{3} - 17 x^{2} + b x - 15$ $C o m p a r i n g c o e f f i c i e n t s :$ $▸ \underset{(i)}{\underset{⏟}{2 c = a}}, \underset{(i i)}{\underset{⏟}{11 c + 2 d} = - 17},$ $\underset{(i i i)}{\underset{⏟}{5 c + 11 d = b}}, \underset{(i v)}{\underset{⏟}{5 d = - 15}}$ $▸ (i v) \Rightarrow d = \frac{- 15}{5} = - 3,$ $(i i) \Rightarrow 11 c + 2 (- 3) = - 17 \Rightarrow c = - 1$ $(i) \Rightarrow a = 2 (- 1) = - 2$ $(i i i) \Rightarrow b = 5 (- 1) + 11 (- 3) = - 38$
	Commented by cortano last updated on 23/Oct/21

	$y e s s i r . i g o t t h e s a m e r e s u l t$
	Commented by cortano last updated on 23/Oct/21

	Answered by Rasheed.Sindhi last updated on 23/Oct/21

	$P (x) = {a x}^{3} - 17 x^{2} + b x - 15$ $\underline{D (x) = 2 x^{2} + 11 x + 5; a = ?, b = ?}$ $▸ T w o r o o t s o f P (x) = 0 a r e a l s o r o o t s$ $o f D (x) :$ $2 x^{2} + 11 x + 5 = 0$ $(x + 5) (2 x + 1) = 0$ $x = - 5 ∣ x = - \frac{1}{2}$ $▸ L e t t h i r d r o o t i s α$ $P (x) = {a x}^{3} - 17 x^{2} + b x - 15$ $= (x + 5) (x + \frac{1}{2}) (x - α)$ $▸ T h i s i s a n i d e n t i t y s o {i t}^{'} s c o r r e c t$ $f o r e v e r y v a l u e o f x :$ $⧫ x = - 5 (L H S = 0 f o r x = - 5)$ $a {(- 5)}^{3} - 17 {(- 5)}^{2} + b (- 5) - 15 = 0$ $- 125 a - 425 - 5 b - 15 = 0$ $25 a + 85 + b + 3 = 0$ $25 a + b = - 88 . . . . . . . . . . . . . . . . . (i)$ $⧫ x = - 1 / 2 (L H S = 0 f o r x = - 1 / 2)$ $a {(- \frac{1}{2})}^{3} - 17 {(- \frac{1}{2})}^{2} + b (- \frac{1}{2}) - 15 = 0$ $- \frac{1}{8} a - \frac{17}{4} - \frac{1}{2} b - 15 = 0$ $a + 34 + 4 b + 120 = 0$ $a + 4 b = - 154 . . . . . . . . . . . . . . . . . . (i i)$ $▸ 4 (i) - (i i) \Rightarrow 99 a = - 352 + 154 = - 198$ $a = - 2$ $b = - 88 - 25 a = - 88 - 25 (- 2) = - 38$ $b = - 38$
	Answered by ajfour last updated on 23/Oct/21

	$2 {a x}^{3} + 11 {a x}^{2} + 5 a x$ $+ 2 {p x}^{2} + 11 p x + 5 p$ $= 2 {a x}^{3} - 34 x^{2} + 2 b x - 30$ $\Rightarrow 11 a + 2 p = - 34$ $5 a + 11 p = 2 b$ $5 p = - 30$ $s o 11 a = - 22 \Rightarrow a = - 2$ $2 b = - 66 - 10 \Rightarrow b = - 38$ $(\underset{\underset{∣ ∣}{-}}{\overset{⌢}{\hat{.} \hat{.}}})$ $(\underset{\sqrt{}}{.} \underset{\sqrt{}}{.}) \int$
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