Solving Polynomial Equations 4: The pattern with the cubic case
Wild Egg Maths
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We begin a careful look at our simplest power series solution to the general cubic equation. Which terms appear? What are the signs in front of them? And what are the coefficients that come with them? These are little puzzles that we must solve to make progress on the bigger problem of finding a general solution.
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hello everyone
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i'm noah wahlberger and thank you for
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joining me
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on this exploration of mathematics
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we're looking at solving polynomial
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equations
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we've introduced a new way of thinking
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about the quadratic case using
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power series rather than radicals
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and we've moved to the cubic case where
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we've gotten some initial
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interesting kind of formulas so we
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started off
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with this as our cubic t plus c one x
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plus c two x squared plus
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c three x cubed so we had replaced the
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earlier
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c zero with t for convenience because
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our orientation at this point is to
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look for a solution which is a power
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series in t
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and we tried x equals the sum of powers
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of t
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up to t to the seven with some
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undetermined coefficients
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a0 up to a7 we saw that one
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of the conditions on a0 was that it was
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actually equal to zero
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and if we looked at that path so a0
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equals zero then we were able to
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uniquely solve for all of the other
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subsequent values of a and here are what
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we got we got a1 equals minus 1 over c1
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a2 equals minus 1 over
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c1 cubed times c2 and so on involving
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powers of c1 actually odd powers of c1
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in the denominator
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that's actually familiar that's the same
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kind of thing that was happening in the
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quadratic situation
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and then we got in the numerator these
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polynomials
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in c
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1 c 2 c 3 these coefficients that are
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appearing here in the original cubic
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along with of course t now we want to
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make sense of this okay so this is um
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an interesting thing that happens in
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mathematics quite a lot where
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we've gotten some some results of a
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computation or maybe a
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series of computations we have some data
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typically in this case a family of
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expressions and we want to see what the
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pattern is
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so that's so important to be able to
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to look at something and to say okay
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what's the pattern here how can we
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you know predict what's going to happen
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if we extend this for example if we went
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a couple of degrees higher say
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introduced a
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sub 8 times t to the eight we would
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expect to get an ace of eight a formula
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for that
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can we predict what the formula for a
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sub eight would be
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okay so it's like a sudoku kind of
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challenge it's a
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a separate little investigation that we
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now have to stop and pause and try to
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figure out
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what the pattern is here okay so what do
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we know
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well we know that the first entries
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in these polynomials here in the
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numerator are the
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catalan numbers okay the 1 1 2
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5 14 42 132 so that part of it is
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is known and that also emboldens us
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because it means that that part of the
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pattern has
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a a nice a nice expression or a nice
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you know formula that that generates it
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and that makes us hopeful that maybe the
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rest of the pattern
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could also be you know explained with
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some
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say pleasant formula now the catalan
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numbers are appearing as the
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coefficients of powers of
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c2 just pure powers of c2
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the other terms which we have now yet to
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explain
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involve also the variable c3
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all of them involve c3 and then also c1
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and some c2s as well so we want to
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figure out two
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kinds of things here let's write this
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down
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so we want to figure out
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what kinds of terms
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appear in one of these numerators
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right like why do we get c1 squared c2
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squared c3 squared and not some
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some other thing why do we get c1 c2 the
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fourth c3 and not some other thing
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why do we execute exactly these three
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things in addition to this power of
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c2 and
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we want to figure out what are
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the coefficients that
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appear with these terms
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in other words why does this term that's
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appearing have a coefficient of 180 in
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front of it what's the significance of
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180
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okay so these are two um great problems
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and
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if you're really keen you might like to
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pause
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the video for an hour or a day
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or a week okay and see if you can
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just from this data here if you can
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figure out
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what the pattern is
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it might be that you know we'll be
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tempted to
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find some more data like to push this up
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a little bit
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to redo the calculation going up some
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more terms so that we actually get more
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damage so we can see
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more of the pattern making it a little
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bit easier to see what's going on
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but let's at first let's sort of see
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what we can
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deal with here okay so what can we say
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about
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about these things here well um
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let's say start with this one this is
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the first one that appears c1
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times c3 okay and then
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this one is c1 c2 c
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3 and then here we have c1 squared c3
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squared
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here we have c1 c2 squared c3
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c1 c2 cubed c3 c1 squared c2 c3 squared
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are we seeing any kind of pattern
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yes we are it appears that
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the terms involved
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in say a
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a sub k are all of the same degree
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the same total degree okay total degree
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namely what is that total degree so a5
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all the terms appear to be a degree four
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namely k minus one so let's just check
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so with a7 all the terms are of degree
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six
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yes they are a6 all the terms are degree
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five
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a three all the terms are degree two a
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two
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all the terms are of degree one
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a one all the terms all in the numerator
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are degree
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zero in other words constant so yeah
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that that
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jives with the data that we've got so
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far
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but not all terms of
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that total degree are appearing so um
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you know that's one possible expression
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in
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in the c's that is of degree four but we
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could also have say
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c1 times c2 times c3 squared
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that's not appearing why is it not
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appearing
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well no now
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you you have to do something that um
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that mathematicians you know routinely
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have to do you have to try to
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stare at this and try different things
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okay you just say okay you assume that
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there is some kind of pattern
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okay that's usually the working
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assumption and you have to sort of see
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well what
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what could possibly be maybe there's
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some relatively simple
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thing that that explains why these terms
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are appearing and not
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the other ones okay so to see what that
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might be
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um let's look at say a six
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okay c to the fifth c one c two
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cubed c three c one squared c
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two c three squared
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so one thing that you might eventually
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come around to realizing is that these
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terms all have
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an additional property namely that if
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you take this
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exponent here this exponent is 5 and you
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look at the index which is 2 the product
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of 5 and 2 is
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- okay if you look over here
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the products of the the indices
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and their exponents are what here
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there's a total of one
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here there's a total of six namely two
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times three and here's a total of
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three because the c is well c three is
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appearing with an
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exponent of one so we're having one
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6 and 3. now 1 6 and 3
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totals to 10 which is the same
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total that we got here so what about
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this one here
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c1 squared so that total would be two
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like if you multiply the degree
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times the index here we have another two
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and here we have a six that's also
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ten so all these terms here have the
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property
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in addition to the one that we've just
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mentioned that
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in addition it appears
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that the
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we have to now introduce this kind of
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new uh concept of degree right that's
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that's sort of what we're being
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what's suggested here that we want to
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define this new concept
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which is not a degree but it's some kind
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of souped-up degree
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that involves the real degrees and also
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the
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the indices okay so let me remind you
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the indices here are are really coming
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from
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from the coefficients here and and where
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they fit in the original
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cubic equations that's c1 is a
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coefficient of x
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2 is the coefficient of x squared c 3 is
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the coefficient of x cubed so
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these these indices here are really are
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quite intimately connected
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with the the algebraic nature of the
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problem they're not just sort of random
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okay so often indices are maybe of sort
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of random you just
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say well let's let this one be a1 and
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this would be a2 and this one be a3 and
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so on
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but that's not what's happening here
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they're actually the indices are closely
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connected with the actual position of
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these objects in the
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the thing so so it makes sense that that
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some formula involving
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the degrees and the indices you know
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might conceivably have some some value
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in it and
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that's what's appearing here so let's
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define that the
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the following quantity
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let's give it a name what should we call
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it okay it involves indices
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and it involves degrees
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let's define the
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index degree how about that okay maybe
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the simplest thing i can think of
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the index degree where is our bold
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uh bold uh
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i think my tool with bold is not around
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here let's
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view it let me get maybe i can just do
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it directly
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how do i make a bold i need another
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toolbar let me just have a look here
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another toolbar
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editing editing sounds reasonable
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let's where's the editing one
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oh it's down here okay i don't know okay
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there it is um i need to put it
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somewhere
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let me put it up here
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so when you first open scientific
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workplace
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these things up here are not populated
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so these are tools and they're
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obtained from from view here
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toolbars you see these toolbars here you
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click that and you have a whole list of
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toolbars
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and the ones that i've checked are the
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ones that have
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appeared and i've been able to put them
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where i want okay you may have to do
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something like i did sort of
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manipulate them a little bit but now
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i've just added the edit
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tool there it is there and that that one
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there it allows me to make something
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bold so i'm going to put b
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for bold and there we get a nice ball so
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i like to
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make my definitions in bold that's a
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relatively simple way of
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denoting the fact that we're defining
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something so define the index degree
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of a term of a term okay now i don't
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want to continue with that ball so let
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me make that
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normal okay so what
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what's the general kind of thing so
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suppose we have say um
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c sub 1 to the
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power i don't know m sub 1
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okay and then something with c2 and
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something with c3
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so we'll change these ones to twos
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and this three c
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sub three super m
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sub three okay so there's a term
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consisting of c ones
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c two c threes we'll define the index
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degree of a term
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thing to be the sum
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of the products of the
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indices and the degrees
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of each of the constituent
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variables
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i.e in this case it would be
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1 times
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m sub 1 plus 2 times
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m sub 2 plus 3 times
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m sub 3.
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okay that's the index degree of this
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term
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and we're claiming in addition appears
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that the following quantity namely that
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thing
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is constant
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is constant
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in each expression
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in each of the ais let's say
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let's check that with the other ones
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just to make sure that's true
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okay so for a3 the index degree here is
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two the index degree here is
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one times one plus one times three sorry
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the index degree here is
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uh two times two that's 4. the index
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degree here is 1
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times 1 and 3 times 1 that's a
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total of 4. yes so the index degree of
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all the terms here is 4.
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the ordinary degree of of both these
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terms is 2.
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the index degree is how about over here
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the index degree is six and here we have
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one plus two plus three which is also
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six
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uh the ordinary degree is three
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uh here the index degree is eight
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two plus six is eight
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one plus four plus three that's also
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eight well the ordinary degree
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is a four
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so it appears let me add this it appears
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then that the
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index degree
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is always exactly twice
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the usual degree
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okay let's carry on not just checking if
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that's true so for a6
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we have degree index degree 10
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index degree 1 plus 6 plus 3 which is
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also 10 and index degree
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2 plus 2 plus 6 which is also 10 which
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is twice the degree which is 5.
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okay and for a7 an index degree of 12
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for this one
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3 plus 9 that's also 12 and here 2
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plus 4 plus 6 which is also 12
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and here 1 plus 8 plus 3 which is also
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12
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and they all have degree otherwise
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6.
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now are we getting
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all such
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we've said that that all the terms that
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are appearing here
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are are of of degree whatever they are
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and then to end of index degree whatever
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they are so for a seven the the total
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degree was
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six
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so let's say so the total degree
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or the the usual degree is
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k minus one while the
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index degree is
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two times k minus one so when
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k is 7 the usual degrees are
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6 of all these terms while the index
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degree is
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2 times 6 which is 12.
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but do all
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terms that have
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this property actually
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appear
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so each of the terms that are appearing
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here has this property that the
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degree is a certain thing and the index
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degree is something but maybe there's
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some other term
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that's not on this list which also has
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this property
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or are these exactly the four
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uh terms that have the property that
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they're
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you know their index degree is six
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and they're sorry their usual degree is
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six and their index degree is 12.
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okay so that's something that we would
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have to kind of check and i'm going to
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leave this as a little exercise so you
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can
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a question is this
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the case
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so um if it is then we're in a kind of a
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happy situation
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because then now we've sort of solved
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half of the problem that we wanted to
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understand we want uh to understand
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which of these terms appears and now we
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have a
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kind of a reasonable you know guess at
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least as to
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how to predict which kinds of terms are
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going to appear
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for one of these a's great
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now the second part of the problem is to
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figure out the coefficients
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that are in front of those terms
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so now we want to understand the
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coefficients
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and perhaps the first thing to do is to
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observe that
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whatever coefficients are happening here
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some of them have
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plus signs and some of them have minus
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signs in front of them so that might be
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a first thing to figure out
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which ones have the plus signs and which
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ones the minus signs
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there's a little bit of inconsistency
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the way we're writing things here
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because there's minus signs in front of
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all these ones except for this one
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so let's just adjust this so it's a
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little bit more in keeping with
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the other things so scientific workplace
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just says some automatic default ways of
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expressing things and sometimes you just
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have to adjust so that
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things are looking a little bit more
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consistent from your point of view so
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now
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things are a little bit more consistent
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and now we can ask okay there's
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these minus signs in front here but say
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in these terms here
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which terms have plus signs
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in front of them which ones have
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negative signs in front of them so how
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about the negative signs
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here there's a negative sign in front of
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c1 and c3
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here also in front of c1 c2 c3
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here in front of c1 c2 squared c3
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here in front of c1 c2 cube c3
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here in front of c1 cubed c3 cubed
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and here in front of c1c to the 4th
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c3
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it appears oops
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it appears that the sign
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of the terms
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is determined by
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well i think it's not too hard to figure
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out what it is at least
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sort of what suggested it's c1 which is
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controlling this
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is determined by um the
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power of c1
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appearing in the term
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uh if there is
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an even power then we get
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well it depends on whether we're
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including this minus sign but if we're
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just including
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just looking at these terms here uh an
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even power then we
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for example this
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this one here is an even power no this
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one here is an even power
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it looks like we get a plus sign
20:53
[Music]
20:55
and with an odd power
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we get a minus sign
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so this is not including the common
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minus sign out front just so we don't
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get confused
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so i think it's reasonably we have a
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reasonable
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guess as to what determines the sign so
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that's also
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making us more confident right there is
21:27
a pattern here it's not just some kind
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of random thing
21:29
it's not like we're talking about prime
21:30
numbers you know some kind of randomness
21:32
now it appears that there's perhaps some
21:34
kind of formula here so now we have to
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get to grips with
21:37
the actual numbers that appear so
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to get to grips
21:44
with the numbers that appear
21:50
as coefficients let's
21:55
first just write them down
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in a nice way so there's a
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kind of an array of of numbers here it's
22:05
kind of a triangular right but the
22:06
these numbers these coefficients are
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getting a little bigger as we go down so
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we're going to use a matrix to
22:14
to do that now one way of doing a matrix
22:16
is to use this sign right here you see
22:18
this symbol here
22:19
so if i pop that up there then i get a
22:22
possibility of creating a matrix of a
22:24
certain dimension
22:25
and that's quite useful but i'm going to
22:27
do it in another way
22:28
i'm going to go to the compute tab here
22:32
and go down to matrices okay
22:35
you see that there's a whole bunch of
22:37
tools for matrices so you can learn a
22:39
lot
22:40
of say linear algebra just by playing
22:42
around with these
22:44
these tools okay but we're going to go
22:46
to fill matrix
22:49
so we're going to create a matrix of a
22:51
certain size
22:52
that's filled with with something okay
22:55
and
22:55
let's fill it with zeros to start with
22:58
just so it's filled with something
22:59
and how big is this thing going to be
23:02
um well we're going down to
23:06
one two three four and then three and
23:08
then
23:09
and this one also has three and then
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this one has two and then this is one
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and one and one so it looks like maybe
23:14
um
23:16
uh well there's seven seven of these so
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seven rows
23:19
okay maybe some might make it seven rows
23:22
and
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uh and it looks like we are going to
23:24
have four columns that might
23:27
might do the trick
23:31
and um okay you can see there there's
23:33
that matrix
23:34
i'm going to just move it actually for
23:36
convenience just temporarily
23:39
i'm going to move it up here somewhere
23:44
so i can manipulate it and fill it in
23:47
with the things here
23:48
i'll put a few spaces
23:52
okay i also prefer to have my matrices
23:55
with
23:56
parentheses so i'm going to go to
23:58
properties this magnifying glasses
24:00
properties so now i can adjust the look
24:02
of this matrix looks a bit more like a
24:05
matrix now to me
24:06
okay so i'm going to populate this with
24:08
the coefficients with just the
24:10
coefficients that are appearing
24:12
okay there's a 1 there's a 1
24:15
and i'm going to ignore the the signs
24:18
two
24:18
and one appearing there and then on the
24:20
next row
24:22
we have a five and another five
24:25
and then a 14 and a three
24:28
and a 21 and then
24:34
42 and 84
24:39
28
24:44
132 a 12
24:48
180 and a 330
24:52
okay that's really capturing the
24:55
the coefficients that appear and the
24:57
first column of this
24:58
we are happy with those are the catalan
25:02
numbers
25:03
now as for these other ones along each
25:06
row the numbers are determined
25:08
but the actual order along the row is a
25:10
little bit random right a scientific
25:12
workplace has just
25:13
spewed out these polynomials in some
25:15
fashion it's not entirely clear
25:17
what you know what the mechanism is but
25:20
we don't have to concern ourselves with
25:21
that we just have to recognize that this
25:22
is not a god-given order so
25:24
it might be that it's convenient to
25:26
rearrange the entries
25:28
along a given row okay so
25:31
let's grab that again and bring it down
25:33
here somewhere
25:36
okay and and now
25:41
for the coefficients
25:47
okay there's that matrix so we want to
25:49
um
25:50
come to grips we want to explain what
25:52
are these things so
25:54
the first column
25:57
first column are catalan numbers
26:02
and i remind you that they are like what
26:04
we call c sub uh
26:07
n and that was fraction 1 over
26:12
n plus 1 times the binomial coefficient
26:16
which is down here
26:18
binomial coefficient 2n choose
26:22
n and if we want to write that out it's
26:25
the same as a
26:26
2n factorial divided by
26:32
n plus 1 factorial times n
26:36
factorial
26:42
okay what are we gonna do well
26:45
we have to stare at these numbers okay
26:47
let's see if we can you know make some
26:49
sense
26:50
of them and
26:54
one natural thing to do is
26:58
to look at the factorization of these
27:02
numbers into
27:03
into primes so that's that's a general
27:06
kind of
27:06
you know idea that you can do you're
27:09
looking for some formula
27:11
if the formula is going to involve
27:13
factorials or
27:14
products of factorials or something like
27:16
that then
27:18
the prime power decomposition of the
27:20
numbers
27:21
is something that's going to help you in
27:23
identifying that
27:25
so the advantage in having them all here
27:27
in a matrix
27:28
is that we can factor them
27:32
we can factor them all at the same time
27:36
and now we are
27:39
kind of happy because look at the
27:41
factorizations even though some of the
27:43
numbers are big
27:44
the prime factors that appear are all
27:48
relatively small in fact the highest
27:50
prime factor is eleven
27:53
two times three times five times eleven
27:55
here we have two squared three squared
27:57
five
27:57
two squared seven three times seven two
28:00
squared 3
28:01
and so on so this is quite a nice
28:05
and now what we're this suggests that a
28:09
formula like this a catalan type formula
28:13
might very well be explaining these
28:16
additional coefficients that are
28:18
appearing here
28:20
so that's good we're happy with that we
28:22
want to find
28:23
such um you know such a
28:26
formula so for that we want to go back
28:29
to our
28:30
our list of actual
28:34
numbers
28:40
and stare at them a little bit more so
28:44
what we want to do is we want to
28:45
associate you know these numbers here
28:47
with these terms
28:49
so we know what the first column so what
28:51
i'm going to do is i'm going to just
28:52
sort of
28:52
put the the relative the relevant um
28:56
pieces of data together so i'm just
28:58
going to put the terms
28:59
that uh that go with the uh the numbers
29:03
uh beside the numbers so we can kind of
29:05
stare at them
29:06
maybe that'll give us some kind of clue
29:08
as to how to get
29:10
the numbers from from the terms
29:15
21 that's that one there um
29:18
the 84 was this one here
29:25
and here
29:28
[Music]
29:38
and
29:50
okay we'll put that right there okay
29:55
so now what we want to do is we want to
29:58
try to explain the numbers that are
29:59
appearing here
30:02
in terms of the terms how can we predict
30:05
that this is going to be 3 times 7 from
30:07
from
30:08
from this so let's let's go down this
30:11
column since we sort of understand this
30:12
column and maybe it's actually helpful
30:14
for us to actually
30:15
put that first column in as well right
30:17
so that there's c2
30:19
and then there's c2 squared
30:24
and c2 cubed
30:28
[Music]
30:29
and c2 to the fourth
30:36
and all the way up to c2 to 6. so now
30:38
we've got all the information so we
30:39
shouldn't disregard this because this is
30:41
you know what we know about so whatever
30:43
connection that
30:45
the 5 has to the c2 cubed
30:48
might give us some insight into you know
30:50
these other ones