Oh, it’s the time for switching the channel into English. Actually, I’m afraid to write in English because I’m blamed not to use English to write blog too often, it may do harm to students. (My English is poor ma, right!) But, firstly, there are less than 10 students reading this blog, so it may be harmless relatively. And next, because of the beautiful fonts, I try to use English in this blog, and that is the only reason ^_^!
F.7C students were just given a quiz, here are some old stuff for your (so-called) enrichment.
Q.1 [Remainder theorem?]
Show that is divisible by .
Q.2 [Binomial theorem?]
Q.3 [Geometry problem?]
Let be the vertices of a regular 2007-gon inscribed in a unit circle. Evaluate .
The questions above may look different to each other, however, by using the roots of unity, we can solve them.
Solution to Q.1
Let be a complex cube root of unity, i.e. and .
If is a polynomial (over ) such that , then is a factor of . Hence is divisible by .
It is easy to check
Hence ; thus is divisible by . Or
for some polynomial
Solution to Q.2
Let be a complex forth root of unity, i.e. and .
The following results are trivial.
if k is a multiple of 4
if k is not a multiple of 4
(where n = 2007)
Put into the above accordingly. We have
Sum up the above 4 equations and use the trivial results just mentioned, we have
Hey, we can simplify it further because we may take , hence
Solution to Q.3
The complex numbers corresponding to the vertices of a regular n-gon inscribed in a unit circle are roots of . Hence, we may let
be the complex numbers corresponding to the vertices of the regular 2007-gon; where .
It is easy to have are roots of the equation
– – – (1)
Suppose we translate the regular 2006-gon to the left by 1 unit, then the vertices will become , and hence
Now, if we can find out an equation whose roots are , then is simply the modulus of the product of roots.
By (1), just set then an equation with roots is
Since the product of roots of the above equation = 2007,
Not enough? If you want to know how powerful in using complex numbers for solving elementary mathematics problems, I suggest an old Chinese popular mathematics for your own leisure reading: 神奇的複數─如何利用複數解中學數學難題
I do admire Mr. Siu’s concept : “I don’t teach: I share". Read his blog to learn better English!