How can I find the roots of a polynomial like x^3-7x+6?

  • Context: High School 
  • Thread starter Thread starter Cetullah
  • Start date Start date
  • Tags Tags
    Roots
Click For Summary

Discussion Overview

The discussion revolves around finding the roots of the polynomial equation x^3 - 7x + 6. Participants explore various methods and approaches for determining the roots, including both informal and formal techniques. The conversation touches on theoretical aspects, practical methods, and the challenges associated with cubic equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about the polynomial's roots and seeks guidance on how to find them, noting that a solution claims the roots are x=1, x=2, and x=-3.
  • Another participant suggests that plugging in simple values like x=0, 1, and -1 can help identify roots quickly.
  • A different participant mentions that there is no straightforward formula like the quadratic formula for cubic equations and recommends graphing the polynomial as a method to find roots.
  • One participant introduces the bisection method as a way to refine root estimates based on graphical analysis.
  • Another participant acknowledges the existence of a general formula for cubic roots but advises against using it due to its complexity, preferring simpler methods instead.
  • A participant highlights the Rational Root Theorem as a tool for identifying possible rational roots based on the factors of the constant term.
  • One participant discusses specific conditions under which certain values can be identified as roots based on the structure of the cubic equation.

Areas of Agreement / Disagreement

Participants generally agree that finding roots of cubic equations can be complex and that simpler methods, such as graphing and testing rational roots, are preferable. However, there is no consensus on a single method, and multiple approaches are discussed.

Contextual Notes

Some methods mentioned depend on specific assumptions about the polynomial's structure and may not apply universally. The discussion also reflects varying levels of comfort with mathematical concepts among participants.

Who May Find This Useful

This discussion may be useful for students studying polynomial equations, educators looking for teaching strategies, and anyone interested in methods for finding roots of cubic polynomials.

Cetullah
Messages
31
Reaction score
0
Hello dear Physics Forums users.

I m currently studying some Integrals from a book which my elder brother studied with years ago, and one of the problems had the denominator:

x^3-7x+6

Well, I m sure that's over my level, and teacher will never ask this, but in the solution it says its roots are x=1, x=2 and x=-3, without showing the steps, or saying how he found them.

Wondering what the steps were, I typed the question at WolframAlpha, and it weirdly said step by step solution is unavailable.

Can somebody show me a formula or a way to find the roots in such a question?
 
Physics news on Phys.org
Cetullah said:
Hello dear Physics Forums users.

I m currently studying some Integrals from a book which my elder brother studied with years ago, and one of the problems had the denominator:

x^3-7x+6

Well, I m sure that's over my level, and teacher will never ask this, but in the solution it says its roots are x=1, x=2 and x=-3, without showing the steps, or saying how he found them.

Wondering what the steps were, I typed the question at WolframAlpha, and it weirdly said step by step solution is unavailable.

Can somebody show me a formula or a way to find the roots in such a question?

In classroom settings, just plugging in obvious numbers often works. You can see right away that x = 1 works. 1 - 7 + 6 = 0. When confronted with a cubic or higher polynomial, always plug in x = 0, 1, -1 first thing.
 
Hmm... I see. Will keep that in mind, thanks :)

It would also be good to know some formal solution if there is one though :)
 
There is no simple equivalent to the quadratic forumula for finding roots of a cubic equation. The sure fire method would be to graph the equation. This is now so easy, using a spreadsheet, that there is no real reason not to do it.

Then look up something called the bisection method of root finding. Using that you can, starting from a rough guess on a graph, refine to any desired number of decimal places.
 
I see...thanks for the answers guys :)
 
I'll add here, just for completeness' sake, that there is a general equation to find the zeros of a cubic. However it's so nasty that it makes you understand why some people don't like math. I'll link to a page that has the formula, but I'll say I don't recommend actually using it. I'd do what Integral suggested, just saying this to expand knowledge.

http://www.math.vanderbilt.edu/~schectex/courses/cubic/
 
First, whenever you look for a root, have the rational root test in your head. Specifically the only integer roots possible are the factors of the constant.

The cubic ##x^3+px+q## has extra structure you can exploit:
  • if ##p + \frac{q}{k} = -k^2## then k is a root,
  • if ##p - \frac{q}{k} = -k^2## then -k is a root.
 

Similar threads

MHB Find Limit of $\displaystyle\frac{\sec x +3}{7x-\tan y}$ at (0,$\dfrac{\pi}{4}$)
  • · Replies 3 ·
Replies
3
Views
1K
MHB Finding roots of this particular polynomial
  • · Replies 4 ·
Replies
4
Views
2K
High School Questions regarding polynomial divisions and their roots
  • · Replies 12 ·
Replies
12
Views
3K
MHB Show that the polynomial has no real roots
  • · Replies 3 ·
Replies
3
Views
3K
High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Root of nth Degree Polynomial f(x) in Spivak Calculus Ch. 7
  • · Replies 7 ·
Replies
7
Views
2K
How to check if a polynomial is irreducible over the rationals
  • · Replies 18 ·
Replies
18
Views
5K
MHB How Can I Prove that the Real Root of a Given Polynomial Equals 3?
  • · Replies 17 ·
Replies
17
Views
4K
High School Can You Prove the Polynomial Has Three Distinct Roots?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Finding the Largest Root of a Polynomial Using Synthetic Division
  • · Replies 1 ·
Replies
1
Views
1K