''Incomplete'' expression

Algebra Level 4

In the expansion of ( 1 − x − k x 2 ) 8 (1 - x- kx^2)^8 , the coefficient of x 3 x^3 is 3.5 times the sum of the coefficients of x x and x 2 x^2 .

k k can be expressed as a b \frac{a}{b} where a and b are coprime positive integers. What is the value of a + b a+b ?


The answer is 5.

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Noel Lo by Noel Lo , 24, Singapore

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1 solution

Noel Lo
Apr 23, 2015

( 1 − x − k x 2 ) 8 (1 - x- kx^2)^8 = ( 1 − ( x + k x 2 ) ) 8 (1 - (x+ kx^2))^8

= 1 - 8 ( x + k x 2 ) (x+ kx^2) + 28 ( x + k x 2 ) 2 (x+ kx^2)^2 - 56 ( x + k x 2 ) 3 (x+ kx^2)^3 + ...

= 1 - 8 x 8x - 8 k x 2 8kx^2 + 28( x 2 + 2 k x 3 + . . . ) x^2 + 2kx^3 +...) - 56 x 3 x^3 +...

= 1 - 8 x 8x + ( 28 − 8 k ) x 2 (28-8k)x^2 + ( 56 k − 56 ) x 3 56k - 56)x^3 + ...

We have 56 k − 56 = 3.5 ( − 8 + 28 − 8 k ) 56k - 56 = 3.5(-8 + 28 -8k)

56 k − 56 = 3.5 ( 20 − 8 k ) 56k - 56 = 3.5(20 - 8k)

56 k − 56 = 70 − 28 k 56k - 56 = 70 - 28k

84 k = 126 84k = 126

k = 126 84 = 3 2 k= \frac{126}{84} = \frac{3}{2}

so a + b = 3 + 2 = 5 a+b= 3 +2 = \boxed{5}

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