In the realm of 6 Dimensions

Algebra Level 5

$\large P(x)=x^6+(x-r)^6+(x-2r)^6-3r$ The polynomial above has exactly two real zeros. Find the value of $r$ such that the sum of the real zeros is $1$ .

Note: Don't forget to try its awesome sister problem . This problem belongs to the set- Questions I've Made ©

The answer is 0.5.

2 solutions

Sanjeet Raria
Mar 10, 2015

We observe that $\large P(r+x)=P(r-x)$ for all $x$ . That means the graph of the polynomial is symmetrical with respect to the line $x=r$ . Further among 2 real zeros, if one zero is $(r+S)$ the corresponding other would be $(r-S)$ . That means the sum of the 2 real zeros will be $2r.$ $\large2r=1 \Rightarrow r=\boxed{.5}$

Awesome problem! @Sanjeet Raria

Had fun solving it!

Harsh Shrivastava - 6 years, 2 months ago

Try its sequel. You'll love that too.

Sanjeet Raria - 6 years, 2 months ago

Thats my way :)

Mvs Saketh - 6 years, 3 months ago

Yes I agree that i first took the vision of the problem straight from your solution . But the concept involved here is very old though. This doesn't belong to a particular individual.

P.S. BTW your solution was really awesome.

Sanjeet Raria - 6 years, 3 months ago

Indeed it belongs only to the mathematicians , i only meant that this perhaps partly came from the solution i posted :)

Mvs Saketh - 6 years, 3 months ago

As soon as I put x-r=t. I knew i would find this comment here :)

Raghav Vaidyanathan - 6 years, 3 months ago

If $r$ is complex the line $x=r$ does not exist.

Joel Tan - 6 years, 3 months ago

I asked to find the value of $r$ . Doesn't it imply that it is real?

Sanjeet Raria - 6 years, 3 months ago

For a complete solution all cases should be considered.

Joel Tan - 6 years, 3 months ago

@Joel Tan – Thanks. See this problem.

Sanjeet Raria - 6 years, 3 months ago

@Joel Tan – We don't need any line. The fact that $P(r-x)=P(r+x)$ is sufficient to get the result. Even when $r$ is an imaginary number. That was for the sake of visual understanding.

Sanjeet Raria - 6 years, 3 months ago

https://brilliant.org/problems/find-the-areaonly-your-logic-can-help-you/?group=3UHxOzwinQpA&ref_id=384997

PLEASE TRY TO DO THIS AWSOME PROBLEM TOO..post a solution if you get......................i am waiting for an awesome solution that i made while creating this problem

Yash Sharma - 6 years, 3 months ago

Niranjan Khanderia
Jan 15, 2018

Solved by graphing trying different values of r.

In the realm of 6 Dimensions

Note: Don't forget to try its awesome sister problem . This problem belongs to the set- Questions I've Made ©

The answer is 0.5.

2 solutions

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