A 2015 degrees! what is it, mercury???

Algebra Level 4

If the roots of the polynomial:

$P(x) =x^{1007}+x^{1006}+x^{1005}\dots\dots+x^2+ x+1$

are $a_1,a_2,a_3,a_4\dots\dots a_{1007},$ what is the value of $\displaystyle{\sum_{i=1}^{1007} F(a_i)}$ when

$F(x)=x^{2015}+x^{2014}+x^{2013}+x^{2012}+\dots\dots+x^2+x+1?$

The answer is 0.

2 solutions

Jon Haussmann
Dec 26, 2014

Note that $P(x) = \frac{x^{1008} - 1}{x - 1},$ so $a_i^{1008} = 1$ for all $i$ .

Then $F(a_i) = \frac{a_i^{2016} - 1}{a_i - 1} = \frac{(a_i^{1008} - 1)(a_i^{1008} + 1)}{a_i - 1} = 0$ for all $i$ , so $\sum_{i = 1}^{1007} F(a_i) = 0.$

Aareyan Manzoor
Dec 24, 2014

first note that $P(a_i)=0$ for $i \in [1,2,3\dots\dots ,1007]\quad$ and that $F(x)=x^{1008} ( x^{1007}+x^{1006}+\dots\dots x+1)+x^{1007}+x^{1006}+\dots\dots x+1$ $F(x)=x^{1008}(P(x))+P(x)$ $F(a_i)=x^{1008}(P(a_i))+P(a_i) =x^{2013}(0)+0=0$ now add it 1012 times $\sum_{i=1}^{1006} F(a_i)=0\times 1007 =\boxed{\LARGE{0}}$

Given that you changed the problem significantly, can you update your solution too? IE there is no $a_ {1012}$ .

I fail to understand what your $F(x)$ in the solution is, and how it corresponds to the polynomial given in the question.

Calvin Lin Staff - 6 years, 5 months ago

sir, $F(x)$ is defined at the end of the question . sorry for the change, there was a silly mistake in the degrees.

Aareyan Manzoor - 6 years, 5 months ago

1) Can you explain how your solution

$F(x)=x^{1008} ( x^{2007}+x^{2006}+\dots\dots x+1)+x^{2007}+x^{2006}+\dots\dots x+1$

corresponds with the problem

$F(x)=x^{2015}+x^{2014}+x^{2013}+x^{2012}+\dots\dots+x^2+x+1$

2) Can you explain how your solution

$F(x)=x^{1008} ( x^{2007}+x^{2006}+\dots\dots x+1)+x^{2007}+x^{2006}+\dots\dots x+1$

is equal to $F(x) = x^{1008} P(x) + P(x)$ ?

Calvin Lin Staff - 6 years, 5 months ago

@Calvin Lin – sorry again, it shoud be 1007 in the solution, not 2007

Aareyan Manzoor - 6 years, 5 months ago

@Aareyan Manzoor – Thank you, this now makes sense to me.

Calvin Lin Staff - 6 years, 5 months ago

A 2015 degrees! what is it, mercury???

The answer is 0.

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