Roots of a 4th degree polynomial

In summary, the problem is to find the roots of a polynomial equation. Taking into account that the equation is x^4-960x^3+91500x^2-6272000x+501760000=f(x), it is advised to use the Newton's Method. Additionally, Ruffini's Method can be used if the polynomial equation is too difficult to factor.
  • #1
Grantismo
4
0
Hi eveveryone I was just hoping for some quick help on frustrating physics related math problem. I won't go into detail on the actual problem becasue i know i found the correct polynomial but i was wondering if there was any easy way to find the roots to this polynomial:

3x^4-960x^3+91500x^2-6272000x+501760000=f(x)
*sorry i haven't figured out how to use latex or w/e it's called*rational roots seems rather arduous with the numbers involved. Any suggestions?(I know there is only one answer about 125.98 i think but i was wondering if there was a way to find an exact answer algebraically or with calculus or something)
 
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  • #2
Apart from using Ruffini's method (rational roots) or Newton's method, both which will require time to yield answers, maybe since this isn't a mathematical problem you can use Mathematica, Matlab, etc... for your solutions.

Also you could have tried Descartes' sign rule, but that wouldn't have helped much anyway.
 
  • #3
rational roots seems rather arduous with the numbers involved.
501760000 isn't a very big number. A computer should be able to factor that before you can blink. It can probably plug every number dividing 501760000 into that polynomial roughly as quickly.


i was wondering if there was a way to find an exact answer algebraically
Is there any reason why you can't simply define r to be a root of that polynomial, and then work with r?
 
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  • #4
I don't think that the roots are rational now that I've looked at a it or a while,
Cyclovenom if you could explain any of those methods i might try them.
 
  • #5
Grantismo said:
I don't think that the roots are rational now that I've looked at a it or a while,
Cyclovenom if you could explain any of those methods i might try them.

Certainly, i will try to answer any questions about the methods, but they are explained in these sites:

Newton's Method

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/approx/Newton.html"

http://www.sosmath.com/calculus/diff/der07/der07.html"

Ruffini's Method

"[URL
 
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  • #6
wow, the Newton method is PERFECT for what I wanted, plus it will also give my teacher a huge laugh (inside joke about approximations)
Thank you SOO much :)
 

1. What is a 4th degree polynomial?

A 4th degree polynomial is a mathematical expression that contains a variable raised to the 4th power and may also include other terms with lower powers. It is also known as a quartic polynomial.

2. How do you find the roots of a 4th degree polynomial?

To find the roots of a 4th degree polynomial, you can use the rational root theorem or the quadratic formula for the first two roots. For the remaining two roots, you can use the cubic formula or the quartic formula, which are more complex and involve complex numbers.

3. What is the significance of finding the roots of a 4th degree polynomial?

Finding the roots of a 4th degree polynomial can help in graphing the polynomial function and understanding its behavior. It can also be used to solve real-world problems involving polynomial equations.

4. Can a 4th degree polynomial have imaginary roots?

Yes, a 4th degree polynomial can have imaginary roots. This is because the quartic formula involves taking the square root of a negative number, which results in complex roots.

5. How do you know if a 4th degree polynomial has real or complex roots?

A 4th degree polynomial with real coefficients will always have either four real roots or two real and two complex roots. The nature of the roots can be determined by analyzing the discriminant of the polynomial equation. If the discriminant is positive, there will be four real roots. If it is negative, there will be two real and two complex roots.

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