Tuesday, October 2, 2007
But it's not in the Standards: Finding imaginary roots, completing the square, factoring and other 'obsolete' topics...
Remember the good old days when students solved 'quadratic-type' equations? Of course, many are still doing this but it is fast becoming a lost art (and some of you may feel it should be!). It is not required in any state math standards or Achieve's, so there's no reason to mention it, right?
Below you will find a 4th-degree (quartic) polynomial equation. The rational root theorem won't help because there are no rational roots. The graphing calculator won't help because there are no real roots! Ok, maybe Mathematica and other Symbolic algebra software could do this, but who exactly programmed this?
Using substitution to rewrite certain 4th degree equations as quadratics (so-called 'biquadratic' equations) used to be covered in some Algebra 2 or advanced classes. Some of you may feel nostalgic about this. However, our challenge today is to solve this by at least TWO 'radically' different methods and then show the solutions are equivalent!
Here's your equation:
x4 + 3x2 + 4 = 0
(a) Explain, without solving, why this equation has no real roots. Should ALL students in Algebra 2 and beyond be able to answer this one?
(b) Solve, by substituting y for x2 and using the quadratic formula. You should eventually arrive at 4 imaginary solutions. This is the way I was taught to solve it, eons ago.
(c) Solve by completing the square and factoring. [Definitely not the first method I would have thought of way back when...]
(d) Show your results are equivalent. This may be annoying! So, which method is easier in your opinion?
(e) Any other method for finding imaginary solutions?
QUICK OPINION POLL
(1) Completing the square (not to mention factoring) is no longer an important topic and should be deemphasized in our curriculum (or omitted).
By the way, is it explicitly mentioned in your state's math standards for Gr 8-12?
(YES NO)
(2) The equation in this post has little relevance to the 21st century and Dave should be ashamed for publishing such trash. Besides, this topic is not included in the Algebra 2 Standards developed by Achieve and ADP.
(YES NO)
You've perhaps assumed that since I've been discussing and complimenting Achieve's standards and the new Algebra 2 End of Course Exam, that I would no longer advocate exposing students to this kind of traditional mechanical 'exercise.'
Well, I taught from the AP Calculus syllabus and I still made time to discuss some ideas and methods that were not 'required'! Further, who exactly will be the ones left on this planet who know how to find imaginary roots for this type of polynomial equation that has no real roots! In case you're wondering, this kind of question has traditionally been taught in Asian countries and still is! (Dave, can you document that? Sure...)
Posted by Dave Marain at 7:02 AM 20 comments
Labels: advanced algebra, algebra 2, completing the square, complex numbers, theory of equations