If (1+ax+bx^2 )(1−2x)^(18) can be expanded using binomial theorem in ascending power of x.Determine the value of a and b,if the coefficient of x^3 and x^(4 ) are both zero.


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Question Number 40523 by scientist last updated on 23/Jul/18

$I f (1 + a x + {b x}^{2}) {(1 - 2 x)}^{18} c a n b e e x p a n d e d u s i n g$ $b i n o m i a l t h e o r e m i n a s c e n d i n g p o w e r o f x . D e t e r m i n e$ $t h e v a l u e o f a a n d b, i f t h e c o e f f i c i e n t o f x^{3} a n d x^{4} a r e b o t h z e r o .$
Commented by tanmay.chaudhury50@gmail.com last updated on 23/Jul/18

$(1 + a x + {b x}^{2}) (1 - 18 C_{1} . 2 x + 18 C_{2} . 4 x^{2} - 18 C_{3} . 8 x^{3}$ $+ 18 C_{4} . 16 x^{4} + . . .)$ $t h e t e r m c o n t a i n i n g x^{4}$ $x^{4} (18 C_{4} . 16 - a . 18 C_{3} . 8 + b . 18 C_{2} . 4)$ $t h e t e r m c o n t a i n i n g x^{3}$ $x^{3} (- 18 C_{3} . 8 + a . 18 C_{2} . 4 - b . 18 C_{1} . 2)$ $i f c o e f f i c i e n t o f x^{3} a n d x^{4} a r e z e r o t h e n$ $a . 18 C_{2} . 4 - b . 18 C_{1} . 2 = 18 C_{3} . 16 i s t e q n$ $a . 18 C_{3} . 8 - b . 18 C_{2} . 4 = 18 C_{4} . 16 2 n d e q n$ $t o s o l v e . . .$ $18 C_{1} = 18 18 C_{2} = \frac{18 \times 17}{2} = 153$ $18 C_{3} = \frac{18 \times 17 \times 16}{3 \times 2} = \frac{153 \times 16}{3} = 816$ $18 C_{4} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2} = \frac{816 \times 15}{4} = 3060$ $i s t e q n$ $612 a - 36 b = 816 \times 16$ $816 \times 8 a - 153 \times 4 b = 3060 \times 16$ $i t s a b i g b i g n u m b e r s o s o l v e p l s . . .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 23/Jul/18
	$(1−18C_1 .2x+18C_2 .(2x)^2 −18C_3 .(2x)^3 +....) terms containing x^3 =−18C_3 (2x)^3 +ax.18C_2 .(2x)^2 +bx^2 .(−18C_1 (2x) =x^3 {−18C_3 .8+a.4.18C_2 +b.−18C_3 .2 } pls post complete question...$
	$(1 - 18 C_{1} . 2 x + 18 C_{2} . {(2 x)}^{2} - 18 C_{3} . {(2 x)}^{3} + . . . .)$ $t e r m s c o n t a i n i n g x^{3}$ $= - 18 C_{3} {(2 x)}^{3} + a x . 18 C_{2} . {(2 x)}^{2} + {b x}^{2} . (- 18 C_{1} (2 x)$ $= x^{3} {- 18 C_{3} . 8 + a . 4.18 C_{2} + b . - 18 C_{3} . 2}$ $p l s p o s t c o m p l e t e q u e s t i o n . . .$
	Commented by MJS last updated on 23/Jul/18

	$also the constant factor of x^{4} is zero, so just$ $continue!$
	Commented by tanmay.chaudhury50@gmail.com last updated on 23/Jul/18

	$o k s i r . . .$
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