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to find coefficient of a particular power of x in a large expression

in the product (x-1)(x-2)(x-3).....(x-1001), find coefficient of x^1000

Note by Kislay Raj
4 years, 3 months ago

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I'am getting the answer as -501501. I worked it out. (Edited after correction from Bob) Saurabh Dubey · 4 years, 3 months ago

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@Saurabh Dubey It's actually -501501. Alternate the sign. Bob Krueger · 4 years, 3 months ago

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@Bob Krueger Oh! I totally forgot that.Thanks for the correction :) Saurabh Dubey · 4 years, 3 months ago

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@Saurabh Dubey Did you use expansion? Aditya Parson · 4 years, 3 months ago

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@Aditya Parson No way!!. You can use the fact that the sum of roots of any equation is equal to the coefficient of the (highest-1)th power divided by the coefficient of the first term. Now, we see in this expression that the roots are (1,2,3,4,....1001). Hence the sum of the roots will be 1+2+3+4....+1001. =((1001)*(1002))/2 = 501501. hence

501501 = (-1)*(coefficient of x^1000)/(coefficient of x^1001) Also we see that the coefficient of the highest power term (x^1001) will be 1. Hence,

501501 = (-1)(Coefficient of x^1000)/1 => coefficient of x^1000 = -501501*. :) Saurabh Dubey · 4 years, 3 months ago

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@Saurabh Dubey Thanks, I didn't know that. But the sum of roots of any equation is that equation? Let us take \(x^2+2x+1\) The sum of it's roots is \((-2).\) But according to your equation it should come as \(2\). So did you mean or forgot to mention absolute value? Aditya Parson · 4 years, 3 months ago

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@Aditya Parson Vieta's fomula's alternate between positive and negative for each coefficient. So his answer is wrong. It should be negative. Bob Krueger · 4 years, 3 months ago

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@Bob Krueger Ya Bob is right. Saurabh Dubey · 4 years, 3 months ago

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