How to find the roots of polynomial of a 5.th order

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Homework Help Overview

The discussion revolves around finding the roots of a polynomial equation of the fourth degree, specifically one with coefficients that are very small and include both real and complex roots. Participants are exploring methods to identify the real roots and clarify the notation used in the results from polynomial solvers.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of a roots function to find the polynomial's roots and question how to interpret the results, particularly regarding the presence of complex roots. There is also inquiry into the meaning of scientific notation as it relates to the roots obtained from different solvers.

Discussion Status

Some participants have provided clarifications regarding the degree of the polynomial and the nature of the roots. There is ongoing exploration of how to handle the results from different polynomial solvers, particularly in understanding scientific notation and its implications for the real values of the roots.

Contextual Notes

Participants note that the original polynomial is of fourth degree, not fifth, and discuss the implications of having both real and complex roots. There is mention of confusion regarding the interpretation of results from different solvers and the notation used in expressing these results.

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Homework Statement


I have a polynimal equation as this
- 0.00000000000049125*T^4 + 0.00000000021358333333333333333333333333333*T^3 + 0.00000290233125*T^2 - 0.032444109375*T + 19.891472013020833333333333333333



Homework Equations





The Attempt at a Solution



I insert those polynomial coeffictions to an line matris
a = [- 0.00000000000049125 0.00000000021358333333333333333333333333333 0.00000290233125 - 0.032444109375 19.891472013020833333333333333333]

a =
-0.0000 0.0000 0.0000 0.0324 19.8915
and call roots function with thi array
roots(a)

and get this values

ans =

1.0e+003 *

4.8369
-1.8781 + 3.0710i
-1.8781 - 3.0710i
-0.6460

I seen, there are real and complex roots. So, How can I find real value of T ??
Should I use a method like bisection ? If so how ?
 
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johny2012 said:

Homework Statement


I have a polynimal equation as this
- 0.00000000000049125*T^4 + 0.00000000021358333333333333333333333333333*T^3 + 0.00000290233125*T^2 - 0.032444109375*T + 19.891472013020833333333333333333

Homework Equations


The Attempt at a Solution



I insert those polynomial coeffictions to an line matris
a = [- 0.00000000000049125 0.00000000021358333333333333333333333333333 0.00000290233125 - 0.032444109375 19.891472013020833333333333333333]

a =
-0.0000 0.0000 0.0000 0.0324 19.8915
and call roots function with thi array
roots(a)

and get this values

ans =

1.0e+003 *

4.8369
-1.8781 + 3.0710i
-1.8781 - 3.0710i
-0.6460

I seen, there are real and complex roots. So, How can I find real value of T ??
Should I use a method like bisection ? If so how ?

First of all, your equation is not 5th "order" (proper term is degree), it's only 4th degree (also called a quartic equation).

There are at most 4 distinct roots to any quartic with rational coefficients (like you have). These roots can be found exactly with certain algebraic operations, but it's fairly tedious. You've apparently already run the equation through a solver and got those roots.

So you have two distinct real roots and two complex roots (the ones with 'i' in them), to give a total of 4 roots. If you only need the real roots, just use those (so T can be either of those values). Just ignore the complex roots unless you have a problem where you need to use those (and you have some means to interpret a complex solution).
 
Thanks for the valuable information.

I tried this online polynomail root solver and entered the polnomial coefficients,
and the these results.

Instead of 1.0e+003 *

4.8369

I get
4836.92716144589

and others are
x2:
-646.021532813122

x3:
-1878.065197692127 + 3071.020155085326i

x4:
-1878.065197692127 - 3071.020155085326i

. I get confused at this point, I can not fully understand "e" concept here, I know it means exponential but I do not know how can I calculate real value of a number which contains that "e" sign.

I mean, in this case
1.0e+003 *

4.8369

is equals to 4836.92716144589 ??

Which numebr should I accepts, can you also help me on this ?
 
Last edited:
johny2012 said:
Thanks for the valuable information.

I tried this online polynomail root solver and entered the polnomial coefficients,
and the these results.

Instead of 1.0e+003 *

4.8369

I get
4836.92716144589

and others are
x2:
-646.021532813122

x3:
-1878.065197692127 + 3071.020155085326i

x4:
-1878.065197692127 - 3071.020155085326i

. I get confused at this point, I can not fully understand "e" concept here, I know it means exponential but how can I calculate real value with that e sign.

I mean
1.0e+003 *

4.8369

is equals to 4836.92716144589 ??

Which numebr should I accepts, can you also help me on this ?

1e3 means 1000 here. The '*' means times (multiply). The first solver was factoring out the 1000, so that the numbers appear smaller. You're meant to take 1000 times each number to get the actual solution.

The second solver just gave the solutions as is.

"x"e"y" also written as xEy or xEXPy (that's why some scientific calcs have that "EXP" key, which means the same thing) where x and y are numbers (generally x is a single digit number greater than or equal to 1 and less than 10 and y can be anything, positive or negative) means x TIMES 10^y.

So 2e4 = 20,000. and 3e-6 = 3/1,000,000.

It's a form of Scientific Notation: http://en.wikipedia.org/wiki/Scientific_notation
 

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