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

#MathProblem

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I'am getting the answer as -501501. I worked it out. (Edited after correction from Bob)

Saurabh Dubey - 8 years, 1 month ago

It's actually -501501. Alternate the sign.

Bob Krueger - 8 years, 1 month ago

Oh! I totally forgot that.Thanks for the correction :)

Saurabh Dubey - 8 years, 1 month ago

Did you use expansion?

Aditya Parson - 8 years, 1 month ago

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 - 8 years, 1 month ago

@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 - 8 years, 1 month ago

@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 - 8 years, 1 month ago

@Bob Krueger – Ya Bob is right.

Saurabh Dubey - 8 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$