Assuming 0 < epsilon << 1 find all the roots of the polynomial epsilon^2x^3+x+1 which are O(1) up to a precision of O(epsilon^2)

Sandra Terrell

Sandra Terrell

Answered question

2022-08-11

Assuming 0 < ϵ 1 find all the roots of the polynomial
ϵ 2 x 3 + x + 1
which are O ( 1 ) up to a precision of O ( ϵ 2 )

Answer & Explanation

Jewel Brooks

Jewel Brooks

Beginner2022-08-12Added 15 answers

First, assume that x ( ϵ ) = O ( 1 ). Then we can say x ( ϵ ) = x 0 + f ( ϵ ), where f ( ϵ ) = o ( 1 ) is some function in ϵ, and x 0 is constant in ϵ. Filling this in, and using that x ( ϵ ) is a root of the equation, we get
ϵ 2 ( x 0 + f ( ϵ ) ) 3 + ( x 0 + f ( ϵ ) ) + 1 = 0.
Expanding the cubic, we get
( ϵ 2 x 0 3 + 3 ϵ 2 x 0 2 f ( ϵ ) + 3 ϵ 2 x 0 f ( ϵ ) 2 + ϵ 2 f ( ϵ ) 3 ) + ( x 0 + f ( ϵ ) ) + 1 = 0.
Now, note that f ( ϵ ) = o ( 1 ), so also f ( ϵ ) 2 = o ( 1 ) and f ( ϵ ) 3 = o ( 1 ). Further note that x 0 = O ( 1 ), so also x 0 2 = O ( 1 ) and x 0 3 = O ( 1 ). So the second, third and fourth terms above are all o ( ϵ 2 ). So if we combine all these o ( ϵ 2 )-terms into one o ( ϵ 2 )-term, we get
ϵ 2 x 0 3 + o ( ϵ 2 ) + x 0 + f ( ϵ ) + 1 = 0.
Rewriting this slightly, to group the terms of the same order, we get
( x 0 + 1 ) + ( f ( ϵ ) + x 0 3 ϵ 2 + o ( ϵ 2 ) ) = 0.
Equating the proper order terms on both sides, we must therefore have x 0 = 1 and f ( ϵ ) = x 0 3 ϵ 2 + o ( ϵ 2 ) = ϵ 2 + o ( ϵ 2 ), from which we may conclude that x ( ϵ ) = 1 + ϵ 2 + o ( ϵ 2 )
If we do not assume x ( ϵ ) = O ( 1 ), then we get more solutions. In particular, the two other solutions are of the form
x ( ϵ ) = ± i ϵ + O ( 1 )
For these solutions, we cannot say that x ( ϵ ) = x 0 + o ( 1 ) for some constant x 0 , so we did not find these solutions with the above method.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?