4.3 Zeros of Polynomials ( The section of useless theorems)

It seems yet again that the wheel has chosen in my favor again and chose me last for blog post...so today in class we learned about the zeros of polynomials...and that this section, told by the legend himself, is full of useless theorems, because of this new technology called a calculator.  I will try to be "bubbly" during this post but there are no promises. There were a few that were of some importance.

Fundamental Theorem of Algebra- If a polynomial f(x) has positive degree and complex coefficients, then f(x) has at least one complex zero.  The first proof of this was done by Carl Friedrich Gauss...btw.

 Going back to stuff that really matters... So for those who don't understand be alarmed because I have no idea either, but luckily the internet had the answer.  This theorem in normal people terms means that for an nth degree polynomial there are n zeros ( I asked Eleni too and this is what she said and sense she is always right I'm going with this).


Next there was the Complete factorization theorem for polynomials that in many words basically says that in a polynomial -- a(x-c1)(x-c2).....(x-ck)... ck is a zero of f(x).



Now to ( again useless but interesting) Descartes' rule of signs....

<---Made by this man

(shout out to Hannah, he has a very spiffy top hat)














I'm not gonna quote the book ( so go read the thing yourself) but it means that for
(+) zeros equal the number of sign variation in f(x) ( the change in sign) or the number minus an even integer
(-) zeros equal the number of sign variations in f(-x) ( the change in sign) or the number minus and even integer.
Both of these are only for real numbers!!!

An example would be

f (x) = x⁵ – x⁴ + 3x³ + 9x² – x + 5               ( take f(x))

          ( count the number of sign changes) 



So in this there would be 4 or 2 positive real zeros.


f (–x) = (–x)⁵ – (–x)⁴ + 3(–x)³ + 9(–x)² – (–x) + 5    ( take f(-x))
= –x⁵ – x⁴ – 3x³ + 9x² + x + 5

                      ( count the number of sign changes)


So in this there would be 1 negative real zero.


And why does this all matter you say?!?! IT DOESN'T!! you see if you know how to graph or use this crazy tool called a calculator, this and many of the other theorems are very unnecessary.   


Next there are theorems about bounds which again don't matter...crazy....



Some of the homework problems had to do with finding a polynomial with prescribed zeros...which is pretty easy...

Find a polynomial f(x) of degree 3 that has the indicated zeros and satisfies the given condition...

Zeros= 2, -1, 3; f(1)= 5

Sense we know that the polynomial has a degree of 3 we know that
f(x)= a(x-2)(x+1)(x-3)

to find a we plug in f(1)=5  then solve for a

5=a(1-2)(1+1)(1-3)
5=a(-1)(2)(-2)
5=a(4)
5/4=a  (don't yell at me Mr.Wilhelm for not using the thingy to make that proper cuz the thingy didn't work)
so f(x)= 5/4(x-2)(x+1)(x-3)   So simply and you get
f(x)= 5/4x^3-5x^2+5/4x+15/2


So that's all that we "learned".... even though it wasn't necessary so some of this blog was a waste of time....but oh well... and if you still don't get math....join the club

Long Live King Thaddeus

And remember kids, teaching chemistry is more fun because in there things explode and you can't do that in math.
Most awesomest video ever 

Jennifer Kendall (5th hour FTW)


And i have lost the game