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|Posted by: bkisker Jan 17 2010, 04:58 PM|
I'm looking for help writing a dynamic question in EV with algorithms to find all roots of a 4th-degree (x^4) polynomial equation.
I'd like to include things like square roots (for example, one of the roots of the equation might be 3 + sqrt(5)) so students can't necessarily just graph the equation and find the roots by looking.
Any help would be greatly appreciated! I can do basic algorithms but I'm not sure my brain can handle creating a question like this one!
|Posted by: McGarrett Jan 19 2010, 01:40 PM|
|If students are expected to find the exact roots of a quartic equation in a classroom situation without the use of a calculator, you will likely need equations from which they can extract two rational roots. This would then allow them to use the quadratic formula to find the remaining roots, whether they are imaginary or real. One method might be to work backwards by creating several quartic equations from their roots. Eg an equation with roots:
x = plus or minus sqrt (5), x = 2, and x = -3 or
x = plus or minus 2i, x = 2, and x = -3 or
x = 3 plus or minus sqrt(5), x = 2, and x = -3
The coefficients of each equation can be stored in lists from which you could randomly select a question – a question that you know will “work” by having “nice” roots. If this is what you are after, let me know and I’ll see what I can do.
|Posted by: McGarrett Jan 24 2010, 02:39 PM|
|I have created a dynamic question that will generate quartic equations with either 4 rational roots, 2 rational and 2 irrational roots, or 2 rational and 2 imaginary roots. Reduced rational answers are shown and any irrational values are expressed in simplest radical form.
These equations should be solvable by students without the use of a graphing calculator. If the coefficients are too large, simply reduce the ranges of b, d, f, and h. The nasty bit was the formatting of the answers. I am certain that there is a much more elegant solution to producing formatted answers, but time constraints won't permit.