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

[Cetullah]

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?

 

[SteveL27]

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.

 

[Cetullah]

Hmm... I see. Will keep that in mind, thanks :)

It would also be good to know some formal solution if there is one though :)

 

[Integral]

There is no simple equivelent 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.

 

[Cetullah]

I see...thanks for the answers guys :)

 

[Vorde]

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/

 

[pwsnafu]

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.

 

[Bohrok]

You can also use the Rational Root Theorem to find the possible rational roots of a polynomial.