Mother of Multiplication!

Algebra Level 4

The coefficient of x 48 x^{48} in the product ( x 2 ) ( x – 4 ) . . . . . . . . ( x – 98 ) (x -2) (x -– 4)........(x -– 98) is


The answer is -2450.

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2 solutions

Karthik Sharma
Sep 3, 2014

use this relation to understand better,

( x a ) ( x b ) = x 2 + ( a b ) x + a b (x-a)(x-b) = x^2 + (a-b)x + ab

( x a ) ( x b ) ( x c ) = x 3 ( a + b + c ) x 2 + ( a b + b c + c a ) x a b c (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+bc+ca)x -abc

( x a 1 ) ( x a 2 ) ( x a 3 ) . . . . . . . . ( x a n ) = x n ( a 1 + a 2 + a 3 + . . . . + a n ) x n 1 + ( a 1 a 2 ) x n 2 . . . . . ( a 1 a 2 . . . a n ) (x-{ a }_{ 1 })(x-{ a }_{ 2 })(x-{ a }_{ 3 })........(x-{ a }_{ n })=x^{ n }-({ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+....+{ a }_{ n })x^{ n-1 }+(\sum { { a }_{ 1 }{ a }_{ 2 } } ){ x }^{ n-2 }-.....-({ a }_{ 1 }{ a }_{ 2 }...{ a }_{ n })

We can find that there are 49 terms in the question, from the AP- 2,4,6,....,98

Using the above relation,

The coefficient of x 48 x^{48} will be-

( 2 + 4 + 6 + . . . . . . + 98 ) = 2450 -(2+4+6+......+98) = \boxed{-2450}

Oh no! I was trying to find the coefficient of 47 all the time....Shizz man!

Pankaj Joshi - 6 years, 8 months ago
Mvs Saketh
Sep 13, 2014

there are 49 terms,, you must jump 48 brackets atleast to get x^(48) , after that you have to jump to constant term of the remaining bracket,, this way u can leave any one bracket and choose x from other brackets in 49 ways,, each giving u every constant term as the coefficient ,, so just sum em up,, and add a negative sign as there are odd number of terms

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