Question

Find the coefficient of x^8 in (x-1)(x-2)(x-3)................(x-10).

Answer 1

We know

(x - a)(x - b) = x2 - (a + b)x + ab

similarly(x - a)(x - b)(x - c) = x³ - (a + b + c)x² + (ab + bc + ac)x

so (x - 1)(x - 2)...(x - 10) = x¹⁰ – (1 + 2 + 3 + . . . . . . . + 10) x⁹ + (1*2 + 2*3 + 3*4 + . . . . . . . + 9*10 + 10*1) x⁸...

so coeff of x⁸ = 2 + 6 + 12 + 20 + 30 .... + 90 + 10 = Σ n(n +1) = Σn2 + Σn + 10

Σ is summation sign. n is 9.

Σn2 + Σn + 10= n(n+1)/2 +n(n+1)(2n+1)/6

= 9(10)/2 + 9(10)(19)/6 + 10

= 9 x 5 + 3 x 5 x 19 + 10

= 135 + 285 + 10

= 430