IIT JEE , AIEEE Forum - Mathematics , Algebra - what is the co-efficient of x^98 in (x-1)(x-2)(x-3)..................(x-98)(x-99)(x-100)?"

KAB

Eg:

Consider (x-1)(x-2)(x-3)

Now expanding it we get x³-(1+2+3)x²+(1*2+2*3+3*1)x-6

Similarly when we expand (x-1)(x-2).......(x-100)

We get x¹⁰⁰-(1+2+3....100)x⁹⁹+(Sum of product of all first 100 integers taken two at a time)x⁹⁸+................

Avinash_Bhat

Hey JOY,

See for example :

(x - 1)(x - 2)(x - 3) = x³ - 6x² + 11x - 6

= x³ - (1 + 2 + 3)x² + (Sum of product of the three integers 1,2,3 taken two at a time)x - (1.2.3)

= x³ - (1 + 2 + 3)x² + (1.2 + 1.3 + 2.3)x - (1.2.3)

SO :

(x - 1)(x - 2)(x - 3) ............. (x - n) = xⁿ - (1 + 2 + 3 + ................ + n)x^n-1 + ( 1.2 + 1.3 + 1.4 + ............. + 1.n + 2.3 + 2.4 + 2.5 + ............ + 2.n + ..................... + (n - 2).(n - 1) + (n - 2).n + (n - 1).n )x^n-2 + ................................ +/- (1.2.3.4. ......... .n)

NOW, LET ME JUST PUT DOWN THE GENERAL RESULTS :

The coefficient of x^n-1 in the expansion of : (x - 1)(x - 2)(x - 3)............(x - n) is : - n(n + 1)/2

The coefficient of x^n-2 in this case is : n(n - 1)(n + 1)(3n + 2) / 24

taruntanuj007

this is a standard rule it works for quadratic biquadratic and for all polynomials it is in theory of equations tooo

ramya_kapoor

OUT OF ALL THE PEOPLE WHO HAVE REPLIED KAB IS THE BEST...............U R VERY RIGHT KAB...

ACTUALLY WAT HAPPENS IS IN A SERIES LIKE THIS WHEN U MULTIPLY ALL THE X...IN ALL THE BRACKETS...U GET X^100...

SO TO GET X^99...U MULTIPLY X OF 99 BRACKETS AND ONE NUMBER FRM EACH...SO U END UP SUMMING 1 TO 100...OK SO NOW WHEN U NEED TO GET X^98............U MULTIPLY X FRM 98 BRACKETS AND THEN MULTIPLY 2 ANY 2 NUMBERS FRM 1 TO 100 AND SUM THEM ALL UP TOGETHER...WAT MISTAKE "CHIRAG" DID WAS HE DIDNT TAKE UP ALL THE POSSIBLE COMBINATIONS OF MULTIPLYING 2 NUMBERS................

OK IF U LIK THE ANSWER PLZ TEL ME..........

ramya_kapoor

NANO...I GUESS ANALYSING THE TREND WUD BE FOOLISHNESS...GO 4 A GUD METHOD LIK MINE AND KAB'S........STUDY MATHS FROM TMH

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