Math Is Fun Forum

It is possible to generalize synthetic division to handle any of these divisions by binomials or larger.  It only involves three steps repeated over and over until the remainder is finished.  That makes these divisions much easier and about 10 times faster since you don't have to write the powers of x.

In fact these divisions can be done by the "usual method" (repeated subtraction algorithm) much faster if one just writes the coefficients of the powers of x.  After all polynomials are a place value system just like base 10 arithmetic.  The main difference is that it is place value in an unknown base x.  As such one cannot carry or borrow.  This makes the arithmetic of polynomials easier than base 10 arithmetic.  And it is not just division that can be done easier in this short form.
Adding, subtracting, multiplying and factoring can be done easier also.

Hi bobbym!
My signature still doesn't show up, does it?  Am I doing something wrong that it won't work?
Where would a thread concerning "the language of mathematics" fit in?  Would we have to
create a new thread to deal with this? 

You are right stefy!  I think I pulled in a "dyslexia" moment.  Humongous anyway, eh? 

Hi 295Ja!  You are most welcome.

Hi to you too stefy!
   I agree it is the human mind that has the problem with these ooncepts of opposite and negative.
"-x" can be read as "the additive inverse of x" or "the opposite of x" both of which are mouthfuls.
"minus x" does not necessarily convey the idea of opposite.  It is more just reading the symbols.
But it is certainly less syllables.  I often read "-x" as "op x" short for "opposite of x".  But it is just
one syllable prefixed to the x and so is easier to say.
   "Negative x" is three syllables prefixed to the x and the word negative brings up the idea of less
than zero and so messes with our minds a bit.  Also it is hard to break the habit of reading "-x"
as "negative x" because it is so heavily ingrained in our language, thought processes and minds.

Many aspiring math students when asked to write the absolute value of "-x" will say it is x; that
is, they will write |-x|=x which is not necessarily true.  They are confusing the "-" for negative
instead of thinking it as saying opposite.  They are so used to "stripping it off" of specific negative
numbers like -2, -3/4, -12, etc. that they just "naturally" want to strip it off of the -x as well.

Sorry for beating a dead horse, but reading "-x" as "negative x" has caused problems for many of
my students over the years.  As such this has been one of my pet peeves.

Have a fantastic day both of you!   

Yes indeedy!  Combinatoris predates computers bunches.  I probably should have said something like "Regarding awareness of multisets, the percent of CS folks that are aware is greater than the percent of mathematicians that are aware."

Ah yes! Generating Functions.  Like x/(x^2-x-1) generates the Fibonacci sequence if you carry the division on forever past the decimal.

And as an attempt to answer your question, stefy, about introducing multisets vs GFs into  math curricula, multisets fit "hand in glove" with basic arithmetic and provide "written manipulatives" for understanding arithmetic.  GFs are a bit further down the road.

Hey bobbym, I am trying with this post another attempt at having a signature.  We will see if it makes it OK. 

And thanks for taking my "LaTex" goof up off the posts!   I wasn't sure how to do it.  Didn't want to mess anything else up. 

Hi stefy!

0/0 is indeed undefined.  That's been mulled over pretty well on one of the threads, eh?
Devante rigged his divisions by binomials and trinomials all to have remainder zero.
That way gives somewhat of a check on whether one is obtaining a correct answer.

One can learn lots of math (a wide variety) by interacting on this site.  And three
cheers for the internet, wikipedia, etc. that gives us almost instant access to lots of info.
There might not be anything left to discover if the biggies like Gauss, Fermat, Euclid, ...
had such communication possibilities.

You are doing fantastic for your age!  Keep it up.  My age?  It is twixt twin primes and has
five prime factors.

Hint:  All those where you are dividing by a binomial or a trinomial have remainder zero.
         Dividing by a binomial yields remainder zero.
         Dividing by a trinomial yields remainder 0x+0 = 0.

Hey Devante,

How about a similar set of exercises involving even and odd fractions?

Does the original problem state that the  marbles are distinguishable or not?

Distinguishability makes a big difference in the way probabilities turn out.  Could it be that the 1/2 comes from indistinguishability and the 2/3 from distinuishability?  The solution in post #5 appears to assume distinguishability.

Some other important concerns in probability:  Repetition vs not,  replacement vs not,  the sample space for the problem.     Probability can be exceedingly tricky!

Computer scientists are way ahead of the mathematicians when it comes to dealing with multisets.  For example Visual Studio 12 has a list of 42 functions that operate on multisets.  However there seems to be quite an analogy between LaTex/PrettyMath and VisualStudio12/PrettyMultisets.  The encoding of Visual Studio 12 is much more complicated than the plain mathematical writing of multisets and their operations.  That seems quite analogous to the complicated coding in LaTex vs its output.  The basics of multiset theory have been known since the 1960's and maybe awhile longer.  The ideas are catching on finally in the mathematical community, but have not been introduced into curricula up through at least high school.

There are some interesting correlations between multisets and prime factors of a number M.  For example If M=12 then the multiset of prime factors of M is PM={2,2,3}.  The number of distinct subsets of PM is always equal to the number of factors of M (FM={1,2,3,4,6,12} has 6 elements).
Also a number M is a perfect square if and only if the number of distinct subsets of PM is odd.
Also HCF(M,N) = X(PMnPM);  LCM (M,N)=X(PMuPN);  MxN=X(PM+PN) where
X and x indicate products, n is for intersection and u is for union, + dumps PM and PN together.
The list is much longer than this, but this is enough to give the flavor of the correlation.

Regarding post #11 about sets:  I vote for doing just sets and not include real analysis.

But I would include formulas about the other half of set theory --- multisets.  Not many folks are aware of multisets.  Sets which allow only one copy of each type of element (typesets) form the set theory that corresponds to two valued (T/F) logic.  Multisets which have at least two of al tleast one type of object (Eg.  {x,x,y,z,z,z,t}, {x,x,y,z}, {x,y,y,y}, etc.) together with typesets forms a bigger set theory in which we have extra operations and relationships.  Eg. addition: {x,y,y}+{y,z,z} = {x,y,y,y,z,z}.  Similarity: A and B are similar if they have the same types of objects but not necessarily the same number of each type.  typeset operator T:  T{x,x,y,z,z,z} = {x,y,z} outputs one of each type of object.   Likeset operator L:  L{x,x,y,z,z,z} = {o,o,o,o,o,o}  changes all objects into one generic type object named "o". 

Of course we still have union (based on maxima) and intersection (based on minima) and equality of sets.  Also there are subsets and differences of sets.

Restricting the types to "o" alone we obtain { }, {o}, {o,o}, {o,o,o}, ... to which we can give the names 0, 1, 2, 3, ...  These fit "hand in glove" with basic whole number arithmetic.  These sets can be used for "written manipulatives".  For example {o,o}+{o,o,o}={o,o,o,o,o} written in terms of their names is 2+3=5.  Inequalities come from the subset relationships.

One might have a bit of trouble finding much about these on the internet.  Most of the work done on these basic concepts has been done by computer sicentists since their programming languages usually include strings that may or may not have multiple copies of some symbols.

I'd be interested to know what you find out about multisets if you are so inclined.

hi bobbym,

In going from the second step of your answer in post #3 to the third step did you miscopy the x^5 as an x^4?  Should dx = ((-1/4)x^5)du?

Teaching:  If you are thinking about teaching in public schools, make sure that maintaining discipline is consistent with your personality.  If not, study diligently about how to do so.  I taught four years in private schools where parents and teachers and administration were on board with making sure their children were polite and respectful.  Still I had trouble keeping them focused on the math at hand.  My personality is just too easy going so I never quite got the hang of holding their attention.  It doesn't matter how well you know the material or how prepared your presentation is, if they are not paying attention it's a bust.  My students learned a good bit but not as much as they could have.  My experience with private schools convinced me that teaching in a public school would not work for me.

Actuary:  The exams needed to become an actuary are from what I have always heard are really comprehensive and difficult.  Grad school in statistics would probably be the route to take in preparation for them.  So that would put you in graduate school for the time being.

Graduate School:  Not a bad idea.  But keep abreast of the requirements for graduation and make sure you progress nicely.  Otherwise it is possible to spend much more time than necessary to get the degree.  If one just enters graduate school and starts taking math courses without focusing on the goal at hand, it is possible to spend lots of time and take lots of courses before exiting the institution.

I knew a high ranking executive at General Telephone who said that he preferred hiring math majors because they could reason and think on their feet.  That's a good qualification for lots of jobs.

Using the search engine "Ask" I put the question "What can I do with a math degree?"  Some of the sites listed are worth looking at.