Demonstrating Cubic Formula to a Student

coconuts
1 ●1 ●1

I'm trying to demonstrate the Lagrange approach to solving a cubic to a student.

w = CyclotomicField(3).gen()
x0 = var('x0')
x1 = var('x1')
x2 = var('x2')
s0 = x0 + x1 + x2
s1 = x0 + x1*w + x2*w^2
s2 = x0 + x1*w^2 + x2*w

After expanding s1^3 + s2^3, the output is cluttered with cube roots of unity (these _should_ cancel out).

I would like to treat w abstractly, where the only relevant constraints are w^2 + w + 1 == 0 and w^3 == 1.

When simplifying, I would like it to collect all coefficients of powers of w and automatically apply the above rules (e.g: w^2 + w == -1).

Or -- is there some other way to do this?

Specifically, at least for the sake of readability, I would like to the roots to be represented only by w.

add a comment

1 Answer

Sort by » oldest newest most voted

answered 10 years ago

tmonteil
27483 ●32 ●205 ●519 http://wiki.sagemath.o...

updated 10 years ago

If you want to work on the symbolic ring (this is what you did with the way to define the variables xi), you should ask for simplification:

sage: (s1^3 + s2^3).full_simplify()
2*x0^3 - 3*x0^2*x1 - 3*x0*x1^2 + 2*x1^3 - 3*(x0 + x1)*x2^2 + 2*x2^3 - 3*(x0^2 - 4*x0*x1 + x1^2)*x2

This works in this case, but you will never be sure how far will the simplification be done. Otherwise, you should define the xi as indeterminates in a well-defined polynomial ring, not as elements of the symbolic ring:

sage: F =  CyclotomicField(3)
sage: w = F.gen()
sage: R = PolynomialRing(F,3,'x') ; R
Multivariate Polynomial Ring in x0, x1, x2 over Cyclotomic Field of order 3 and degree 2
sage: R.inject_variables()
Defining x0, x1, x2
sage: s0 = x0 + x1 + x2
sage: s1 = x0 + x1*w + x2*w^2
sage: s2 = x0 + x1*w^2 + x2*w
sage: s1^3 + s2^3
2*x0^3 - 3*x0^2*x1 - 3*x0*x1^2 + 2*x1^3 - 3*x0^2*x2 + 12*x0*x1*x2 - 3*x1^2*x2 - 3*x0*x2^2 - 3*x1*x2^2 + 2*x2^3

link

Comments

Thanks for the help! As you can see, I'm quite new at this. Is there anywhere I can get a "birds eye view" of the whole API? The documentation seems cluttered with test cases...

coconuts ( 10 years ago )

add a comment

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 10 years ago

Seen: 300 times

Last updated: May 11 '15

Demonstrating Cubic Formula to a Student savecancel