We used the method where you first convert the cubic into a depressed cubic. A depressed cubic is missing the term. It looks like this:
You convert a normal cubic into a depressed cubic with a substitution. Starting with a general cubic polynomial:
Make the substitution:
And you end up with the depressed cubic:
Finally you find the roots for the depressed cubic to solve for and substitute back to find the roots for the original equation. Solving for the depressed cubic is a multi-step operation, but it’s not too difficult. The above link explains it nicely.
To make it easier, we decided to try to solve a cubic that was already depressed. We picked the following because one of the three roots () is obvious.
The other two roots are easy to discover. Since is a root, we know can be expressed where P is a quadratic expression. Long division yields which tells us the other two roots are .
So now we know what to expect.
When you solve this depressed quadratic you get:
If you put it thru a calculator, you’ll find the above expression indeed equals 1. I was a little surprised to find two irrationals separated by an integer in a non-obvious way. I also found it unsatisfying, since I could not simplify this expression all the way to 1 using algebra.
Let’s look at another cubic solution. Wikipedia tells us the following formula will give us the real root:
To solve we plug , , , and into the above to get:
which simplifies to
and finally evaluates (with a calculator) to:
Although that’s interesting, it’s no more satisfying than the first solution. I feel I’m missing something since I can’t take it all the way to algebraically. It was still a fun exercise though. I’ve occasionally wondered about the methods for solving cubic equations, and now I’ve finally seen it done.
I can see why they don’t teach this in high school. It’s too complicated to ask people to memorize, so they wouldn’t want it on a test. And it gives funny looking results, full of square- and cube-roots, that are difficult to manipulate.
Hmmm, now I’m wondering about a general method for finding the roots of a 4th-order equation. Or even a for a 5th-order equation.
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