Algebraic Pursuit

Algebra Level 3

What is the sum of all coefficients of the polynomial below?

p ( x ) = ∏ n = − 10 10 ( x + n ) p(x)=\prod_{n=-10}^{10} (x+n)

For example, the sum of coefficients of a polynomial q ( x ) = a 1 x k + a 2 x k − 1 ⋯ + a k + 1 q(x)=a_1 x^k+a_2x^{k-1}\cdots+a_{k+1} is a 1 + a 2 + ⋯ + a k + 1 a_1+a_2+\cdots+a_{k+1} .


The answer is 0.

Hamza A by Hamza A , 19, USA

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1 solution

Aareyan Manzoor
Jul 26, 2019

for a polynomial , the sum of its coefficient is the polynomial evaluated at 1, i.e: q ( x ) = a 1 x k + a 2 x k − 1 ⋯ + a k + 1 → q ( 1 ) = a 1 + a 2 + ⋯ + a k + 1 q(x)=a_1 x^k+a_2x^{k-1}\cdots+a_{k+1}\to q(1)=a_1+a_2+\cdots+a_{k+1} hence we are looking for p ( 1 ) = ∏ n = − 10 10 ( 1 + n ) = 0 p(1)= \prod_{n=-10}^{10} (1+n) =\boxed{0} since the product hits n = − 1 n=-1 .

7 Helpful 1 Interesting 4 Brilliant 0 Confused

Yes, this question is very trivial when you realize that a 1 + a 2 + ⋯ + a k + 1 a_1+a_2+\cdots + a_{k+1} , the sum of the coefficients.

hi bye - 1 year, 10 months ago

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