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born in 1946. Retired now, but I still enjoy watching your channel every morning while I have my coffee. Often I will pause the video and attempt to solve the problem myself, but then I expect most of your viewers do that from time to time.
@@Pope_Balenciaga I've been around computers since the early 1960s. Most people my generation were not exposed to computing or only in later life - but if you worked in some areas of mathematics or physics then you had the opportunity.
30 years on the lower side of 1983. As a mathematical physicist (and teacher), I really appreciate your old-fashion blackboard style. I find myself stopping your videos and trying to do the problems myself, and then watch your more elegant solution. My significant other finds it amusing that I view your videos as relaxing entertainment. Thanks.
If you let f(x) to be the cubic in the question then f(x+1) has roots alpha-1,beta-1 and gamma-1. Then these new roots satisfy this new cubic so you can find the sum of cubes in terms of sums of squares and their linear sum
@@anshumanagrawal346 bro in which grade are you? And what're you planning to do in future? I'm also from India just asking for some suggestions, I'm in 10th grade and hella interested in maths and physics...
Born in 2005, haven't yet finished last grade. Love your videos. Due to my lack of expirience or knowledge sometimes I do have to watch some parts again and sometimes stop the video to understand what's been done, but at the end I understand and that's what matters haha
Born in 1974. Didn't like math till 1986, when a math teacher taught us to think and not to just apply rules. Ended up in engineering, but on signal processing, an highly math inclined area!
Yeah. I hated math in high school, I failed and I probably only slipped by because of covid. But my computer science program had a single calculus class in it. I thought it would be good to go through my math textbook from high school while I was on my break to catch up or better yet, get ahead. But last year to my surprise I actually loved going through the textbook, and I'm guessing it's because I actually got to see the derivations and proofs behind the theorems, rather than merely the rule I'll be using in the exam. Now I like math textbooks and lectures, currently doing multivariable calculus and Concrete Mathematics.
Born in 2012, expert in data science, cryptography, relativity theory and homotopy theory as well as having worked for years in the industry, I still love mathematics.
@@rudrapratapsingh3976 He means u don't need to expand the equation (alpha-1)^3+(beta-1)^3+(gamma-1)^3=0 Just change alpha^3=.....to (alpha-1)^3=..., (beta)^3=... to (beta-1)^3..., and gamma^3=... to (gamma-1)^3=.... And let the sum of them equal to zero
I was in graduate school for the second time in 1983. I'm so old, I might not live through this video. I loved grad school. Only grad school, two kids, 2 jobs, and 2 houses forced me to stop.
In 1983 I had already finished 10 calculus classes, every calculus class available at my university, ending with a class in partial differential equations. I get great value from and have deep respect for the content Michael provides and the breadth of topics he covers.
Apparently you are or almost are 60 years old. However how do we calculate 10 calculus courses? I only estimate 8 including pre-calculus and calculus in high school.
@@roberttelarket4934 I didn't count any high school classes, although I learned a little calculus in high school. I also didn't count any Engineering classes, which used a lot of calculus. One class was advanced ordinary differential equations that was heavy on using Green's function solutions. I counted calculus one even though I placed out of it after three weeks. I also audited classes in number theory and chaos. I was very good at numerical analysis, but I didn't count that class either despite solving differential equations numerically.
I was highly amused by your opening request about which side of 1983 we were born on, when I reflected that, in that year, I was possibly older than you are now. Probably not, but it was definitely an amusing thought. I had graduated college, got drafted into the military, served my term & got discharged, entered graduate school, dropped out after several years, and was several more years into my first regular job by then. I know. TMI. I would only add that I like your videos, because you don't tread on heavily-traveled ground; there's always something new & challenging in each episode. I encourage you to keep that up! Fred
I was 42 years old in 1983. I'll let you do the maths. Inspired by Kasner and Newman and Eric Temple Bell (still got them on my shelf) but latterly Abramowitz and Stegun.
Born in 1960, got a BS in math and still love it, and used math moderately often throughout my 40-year programming career to solve some tricky problems.
Michael, the system at 3:55 is equivalent to the system at 4:53 therefore only three equations that cannot give a solution for the three roots and a. It is because you introduce the fourth equation at 6:39 that the system of four unknowns becomes solvable.
1983? I had graduated college, had spent a few years in industry, and was applying to PhD programs. (So I don't think I'm in your typical audience demographic.) I envy your climbing - at my age I can't manage the stuff you do. (I still get out and do some 'technical hiking' - off-trail with some class-4 stuff.)
In 1983, I was demoted from the top-level 3rd-grade math group to the middle group because I wasn't doing my homework. I was crushed! I eventually worked my way back up, joined the math team in high school, did a ton of competitions & stuff, and am still a happy user of mathematics to this day.
this video covers a problem that could have been assigned for extra credit in the algebra class I took in 1983 I watch your videos while doing housework as a way to keep my mind fresh on these topics
Thanks Michael. Born in 1961 and i have always liked maths. I have been following you since the very beginning and I have learned a lot. Now I want you to show us how to calculate α, β and γ
1956. Had a love/hate relationship with math until my junior year in HS when our math teacher "Turkey" Thompson explained that a) mathematics was a language for expressing and reasoning about abstract ideas; b) that the most important word in that language was "equals"; c) that equations were sentences that could be read by substituting the phrase "even though they may look different, [the thing on the left] is THE EXACT SAME THING as [the thing on the right]"; d) that the variables of algebra could be read as the phrases "how much of some thing we choose to call y" and "this much of some thing we choose to call x" because itʻs smart to be lazy and use one symbol rather than whole a bunch of words to express that idea; e) and that because the two sides of an equation were the exact same thing, that sameness relationship didnʻt change if we did the exact same thing to both sides, allowing us to reason out complex problems by "using clever sequences of simple steps" to take advantage of the invariance of that sameness property. The day those ideas clicked, I fell in love with math. Mr. Thompson also made the concept of a limit intuitive for me by using the example of eating half of a pie, then half of what was left, again, and again, and so on. He then proved that no matter how tiny a slice of remaining pie I could imagine, I could always repeat this operation enough times to get an even smaller slice. When I saw the epsilon-delta definition of a limit for the first time, I struggled for a couple of days to get my head around it, as almost every calculus student does, then suddenly realized "Oh, this just a continuous generalization of the pie-eating thing!" I literally think in mathematics these days.
Born 22 years after this competition... As an aside, I've noticed that the earlier math olympiad questions seem easier the farther back you go. It seems like the difficulty of questions is getting harder year by year. No math olympiad would put this question today, for instance
Yeah. This would be considered pretty standard by JEE standards in my country. I was simply commenting on the fact that the so-called standard questions of today were olympiad questions forty years ago
@6:45 Observing you generate the mathematical expression helps student understand how you arrived to (deducted) the current state of expression. I would like to make a suggestion and that is to avoid allowing your torso from obstructing the student's view of you writing the expression. Maybe standing with shoulders aligned perpendicularly to the plane of board and, as you write, stepping backwards --- like a sword fencer where your chalk is your sword.
There is a very straightforward way to do this that requires only 3 matrix multiplications and very little thinking. Set A to be the companion matrix of X^3-6X²+aX+a. A is trigonalizable, and the values on its diagonal will be alpha, beta, gamma. Because of this, A-1 is trigonalizable as well and the values on the diagonal are alpha-1, beta-1, gamma-1. You can then compute (alpha-1)^n + (beta-1)^n + (gamma-1)^n by computing the trace of the matrix (A-1)^n. The case n=3 is pretty straightforward and explicit provided you don't mess up your calculations. It's not necessarily faster than the method presented in the video but it has the advantage of being very straightforward and providing an "easy" way to compute the sum of the n-th powers of the roots (or the roots - t) of an explicitly given polynomial in terms of its coefficients. It's not surprising that such an expression exists, Galois theory / the theory of symmetric polynomials tell us to expect it, but this provides a kind of efficient way of computing it. It is most notable if you're considering a domain R that is not integrally closed - you know that for any monic polynomial with coefficients in R, the sum of the n-th powers of the roots of such a polynomial will be an element of R (which again is true because of the fundamental theorem of symmetric polynomials but I really like how concise the proof is).
@@chessematics Personally my 12th century mongolian ears cannot find any pleasure except with the primal sounds of guttural throat singing. chingis saya sэээээээээээ odo ba rangeh ne seeeeeeee
I was so bored taking freshman algebra that I was teaching myself calculus on the side with my high school library books in 1984. So I guess I’ve had a thing for extracurricular math for a while. I should check if you have anything on Clifford or geometric algebras. A little obsessed by those now.
Heh, I still have my high school math textbook from 1983 -- Algebra and Trigonometry 2, Dolciani, Wooten, Beckenbach,Sharron, Houghton-Mifflin 1971 ...
I was born in 1960. I'm glad I'm old enough to remember what it was like before the internet, and compare it to what it is like afterwards. Those who have known nothing but the internet would do well to examine at the world before. Anyway, the infinite of maths available on the internet keeps me more than happy, as does this channel.
Born in 2003, absolutely love mathematics and I am hoping to become a theoretical physicist. Currently working through a book called tensors manifolds and forms, so your abstract algebra videos are a great way to wind down for me. thanks for the great content
@@Pope_Balenciaga it’s in my course but In three years time when I study general relativity. I however love general mathematical structures as it often gives a deeper understanding to the framework of the model of physics we use. I also can’t wait for three years to study gr, so I’m studying it after I’ve finished that book haha
In 1983 I was in uni, studying number theory, data structures, and stuff; a little more than a decade earlier I figured out how to compute log[b](a) as a continued fraction.
Very nice calculations and usage of the blackboard with colours. I could follow all these steps, knowing the Vieta laws, binomic and trinomic formula etc. But I was wondering if the exercise from Austria has any deeper sense. Has the result a special meaning, or is it just an exercise for algebraic methods?
Essentially writing the sum of cubes of the roots of a cubic polynomial in terms of the sum of its squares which may be obtained by expanding the square of the sum of the three roots.
Nice video! It seems that there has been some effort lately to make prettier miniatures for the videos. That's great, but it has the annoying disadvantage for hiding the problem. I'd prefer a miniature that gives me an idea of the content of the video before clicking
Michael Penn stated at the end that x^3-6x^2+(41/5)x+41/5 = 0 has all complex solutions. Not true. A cubic equation has to have at least one real root.
This is a much nicer solution than my first thought, which was to use the cubic formula to get gnarly expressions for the roots in terms of a, plug into that equation relating the roots, and solve the resulting gnarly equation in a. If anyone likes pain they could try that. I'd be interested to know if there's a reason it couldn't work, just not enough to try it myself.
I used Vieta's formula to get a system of equations for the roots then used the cube of difference formula 3 times for the given equation. Using a;gebraic machinations I was able to get a equation for a and solve it. Got a = 123/15 in the end without finding every root individually
Let M be the matrix [ 0 1 0 ] [ 0 0 1 ] [-a -a 6] The characteristic polynomial of M is the polynomial that you have. The whole point is to use the equation Tr( (M-I)^3)=0. You only need that Tr(M)=6 Tr(M^2)=36-2a and you will get an equation in a, without even touching alpha, beta, gamma.
You could do this more mechanically by just stating the condition for alpha, beta, and gamma in terms of elementary symmetric polynomials which are also equal to coefficients of the polynomial up to a sign. That's more work, though.
I was born in 1973, which is cool because it is a prime number (in the last 100 years these years are also prime : 1931, 1933, 1949, 1951, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017)
I've been putting these questions in chatgpt and it solved this one instantly, I wonder if it's because someone else did or it's just really good at algebra. EDIT: now that I recall it doesn't use user inputs as training sets, it's just good at algebra, it knows vieta's formula and manipulates things like wolfram does. It's quite nice.
I was born before 1983. Then again, I usually skip through the videos because I already know most of this stuff and have worked with very advanced mathematics. So, don't let me skew your demographics.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
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Who's formula?
brilliant is pretty crap ngl
only the articles are okay
All cubic formulas with real coefficients have at least one real root. Can you please at least tell us the real root?
Born 1957 and still loving mathematics.
Not alone !!!
Born 1956 ditto.
1955 and still going.
Same here!
Respect
born in 1946. Retired now, but I still enjoy watching your channel every morning while I have my coffee. Often I will pause the video and attempt to solve the problem myself, but then I expect most of your viewers do that from time to time.
Gotcha by 8 years. 1938 here.
You guys are legends. Hope I'm in touch with technology in my old age
@@Pope_Balenciaga I've been around computers since the early 1960s. Most people my generation were not exposed to computing or only in later life - but if you worked in some areas of mathematics or physics then you had the opportunity.
I have the same routine, but I was born 46 years after you.
30 years on the lower side of 1983. As a mathematical physicist (and teacher), I really appreciate your old-fashion blackboard style. I find myself stopping your videos and trying to do the problems myself, and then watch your more elegant solution. My significant other finds it amusing that I view your videos as relaxing entertainment. Thanks.
If you let f(x) to be the cubic in the question then f(x+1) has roots alpha-1,beta-1 and gamma-1. Then these new roots satisfy this new cubic so you can find the sum of cubes in terms of sums of squares and their linear sum
Exactly!
Yep, thats what I did.
This is the beautiful solution. 😊❤
same. Then I just used newton's sums.
@@anshumanagrawal346 bro in which grade are you? And what're you planning to do in future? I'm also from India just asking for some suggestions, I'm in 10th grade and hella interested in maths and physics...
Born in 2005, haven't yet finished last grade.
Love your videos. Due to my lack of expirience or knowledge sometimes I do have to watch some parts again and sometimes stop the video to understand what's been done, but at the end I understand and that's what matters haha
That's when you learn!
Born in 1974. Didn't like math till 1986, when a math teacher taught us to think and not to just apply rules. Ended up in engineering, but on signal processing, an highly math inclined area!
@rafaelgcpp it's worth to mention name of your maths teacher.
For real, i think most people hates math because the school methodology sucks...
Yeah. I hated math in high school, I failed and I probably only slipped by because of covid. But my computer science program had a single calculus class in it. I thought it would be good to go through my math textbook from high school while I was on my break to catch up or better yet, get ahead.
But last year to my surprise I actually loved going through the textbook, and I'm guessing it's because I actually got to see the derivations and proofs behind the theorems, rather than merely the rule I'll be using in the exam.
Now I like math textbooks and lectures, currently doing multivariable calculus and Concrete Mathematics.
Born in 2012, expert in data science, cryptography, relativity theory and homotopy theory as well as having worked for years in the industry, I still love mathematics.
So, you are between 10-years-old and 11-years-old, as you were born in 2012.
"Still love mathematics" 😂
Replace x by (x+1)
The roots of the polynomial now becomes alpha-1 beta-1 and gamma-1
Then use a^3+b^3+c^3-3abc formula
Pardon, could u pls explain the last step?
@@rudrapratapsingh3976
He means u don't need to expand the equation (alpha-1)^3+(beta-1)^3+(gamma-1)^3=0
Just change alpha^3=.....to (alpha-1)^3=..., (beta)^3=... to (beta-1)^3..., and gamma^3=... to (gamma-1)^3=.... And let the sum of them equal to zero
There had to be a shorter answer for a contest or exam situation.
I was in graduate school for the second time in 1983. I'm so old, I might not live through this video. I loved grad school. Only grad school, two kids, 2 jobs, and 2 houses forced me to stop.
In 1983 I had already finished 10 calculus classes, every calculus class available at my university, ending with a class in partial differential equations. I get great value from and have deep respect for the content Michael provides and the breadth of topics he covers.
Apparently you are or almost are 60 years old. However how do we calculate 10 calculus courses? I only estimate 8 including pre-calculus and calculus in high school.
@@roberttelarket4934 I didn't count any high school classes, although I learned a little calculus in high school. I also didn't count any Engineering classes, which used a lot of calculus. One class was advanced ordinary differential equations that was heavy on using Green's function solutions. I counted calculus one even though I placed out of it after three weeks. I also audited classes in number theory and chaos. I was very good at numerical analysis, but I didn't count that class either despite solving differential equations numerically.
I was highly amused by your opening request about which side of 1983 we were born on, when I reflected that, in that year, I was possibly older than you are now.
Probably not, but it was definitely an amusing thought.
I had graduated college, got drafted into the military, served my term & got discharged, entered graduate school, dropped out after several years, and was several more years into my first regular job by then.
I know. TMI.
I would only add that I like your videos, because you don't tread on heavily-traveled ground; there's always something new & challenging in each episode.
I encourage you to keep that up!
Fred
I was 42 years old in 1983. I'll let you do the maths. Inspired by Kasner and Newman and Eric Temple Bell (still got them on my shelf) but latterly Abramowitz and Stegun.
I was born in 1977. I still love math. I ended up going into theoretical/computational physics, but I still do “fun” math when I can.
Born in 1960, got a BS in math and still love it, and used math moderately often throughout my 40-year programming career to solve some tricky problems.
Michael, the system at 3:55 is equivalent to the system at 4:53 therefore only three equations that cannot give a solution for the three roots and a. It is because you introduce the fourth equation at 6:39 that the system of four unknowns becomes solvable.
he might have done that for the simpler calculations but I'm not sure
@@landy4497 Yes that's right it is useful for the calculations to have the two forms.
Your new title slides are hilarious. And in 1983 I was doing high school math contests...
Finally someone stands up to the hegemony of Big Polynomial
1983? I had graduated college, had spent a few years in industry, and was applying to PhD programs. (So I don't think I'm in your typical audience demographic.)
I envy your climbing - at my age I can't manage the stuff you do. (I still get out and do some 'technical hiking' - off-trail with some class-4 stuff.)
In 1983, I was demoted from the top-level 3rd-grade math group to the middle group because I wasn't doing my homework. I was crushed! I eventually worked my way back up, joined the math team in high school, did a ton of competitions & stuff, and am still a happy user of mathematics to this day.
Born 2007 and am shocked how few of my peers are here
this video covers a problem that could have been assigned for extra credit in the algebra class I took in 1983
I watch your videos while doing housework as a way to keep my mind fresh on these topics
Look at this lighting! Jesus! Thats real quality right there!!!
Thanks Michael. Born in 1961 and i have always liked maths. I have been following you since the very beginning and I have learned a lot.
Now I want you to show us how to calculate α, β and γ
1966, so I’m ancient for this channel.
Great problem, professor. Thank you.
Not sure about that. I did my PhD in 1960.
1956. Had a love/hate relationship with math until my junior year in HS when our math teacher "Turkey" Thompson explained that a) mathematics was a language for expressing and reasoning about abstract ideas; b) that the most important word in that language was "equals"; c) that equations were sentences that could be read by substituting the phrase "even though they may look different, [the thing on the left] is THE EXACT SAME THING as [the thing on the right]"; d) that the variables of algebra could be read as the phrases "how much of some thing we choose to call y" and "this much of some thing we choose to call x" because itʻs smart to be lazy and use one symbol rather than whole a bunch of words to express that idea; e) and that because the two sides of an equation were the exact same thing, that sameness relationship didnʻt change if we did the exact same thing to both sides, allowing us to reason out complex problems by "using clever sequences of simple steps" to take advantage of the invariance of that sameness property.
The day those ideas clicked, I fell in love with math.
Mr. Thompson also made the concept of a limit intuitive for me by using the example of eating half of a pie, then half of what was left, again, and again, and so on. He then proved that no matter how tiny a slice of remaining pie I could imagine, I could always repeat this operation enough times to get an even smaller slice. When I saw the epsilon-delta definition of a limit for the first time, I struggled for a couple of days to get my head around it, as almost every calculus student does, then suddenly realized "Oh, this just a continuous generalization of the pie-eating thing!"
I literally think in mathematics these days.
In my prime at 23, studying theoretical physics, rowing at Orca.
I can’t help but find humor in the fact you used a prime number to define your “prime” age.
-Stephanie
MP Editor
Born 22 years after this competition...
As an aside, I've noticed that the earlier math olympiad questions seem easier the farther back you go. It seems like the difficulty of questions is getting harder year by year. No math olympiad would put this question today, for instance
This would definitely be considered routine nowadays.
We solve these exact questions in alevel further mathematics
Yeah. This would be considered pretty standard by JEE standards in my country. I was simply commenting on the fact that the so-called standard questions of today were olympiad questions forty years ago
@@__8474we do indeed. :)
@@ribhuhooja3137 Agreed. It's a 6-line solution using standard polynomial properties.
7:07 lol I wonder if that “like” animation was put right then intentionally
@6:45 Observing you generate the mathematical expression helps student understand how you arrived to (deducted) the current state of expression. I would like to make a suggestion and that is to avoid allowing your torso from obstructing the student's view of you writing the expression. Maybe standing with shoulders aligned perpendicularly to the plane of board and, as you write, stepping backwards --- like a sword fencer where your chalk is your sword.
Thank you! Your videos are awesome and engaging.
Glad you like them!
There is a very straightforward way to do this that requires only 3 matrix multiplications and very little thinking.
Set A to be the companion matrix of X^3-6X²+aX+a. A is trigonalizable, and the values on its diagonal will be alpha, beta, gamma. Because of this, A-1 is trigonalizable as well and the values on the diagonal are alpha-1, beta-1, gamma-1. You can then compute (alpha-1)^n + (beta-1)^n + (gamma-1)^n by computing the trace of the matrix (A-1)^n. The case n=3 is pretty straightforward and explicit provided you don't mess up your calculations.
It's not necessarily faster than the method presented in the video but it has the advantage of being very straightforward and providing an "easy" way to compute the sum of the n-th powers of the roots (or the roots - t) of an explicitly given polynomial in terms of its coefficients.
It's not surprising that such an expression exists, Galois theory / the theory of symmetric polynomials tell us to expect it, but this provides a kind of efficient way of computing it.
It is most notable if you're considering a domain R that is not integrally closed - you know that for any monic polynomial with coefficients in R, the sum of the n-th powers of the roots of such a polynomial will be an element of R (which again is true because of the fundamental theorem of symmetric polynomials but I really like how concise the proof is).
Born in 1958. Still interested in mathematics. Enjoy the channel and still climbing
Born 1979. The music was better then. I teach math now, which means I can count watching these videos as Continuing Professional Development.
Born 1824. Beethoven just premiered his Ninth. Whole romantic age is coming. I'm at heaven.
Good for you!
@@chessematics Personally my 12th century mongolian ears cannot find any pleasure except with the primal sounds of guttural throat singing.
chingis saya sэээээээээээ odo ba rangeh ne seeeeeeee
I was so bored taking freshman algebra that I was teaching myself calculus on the side with my high school library books in 1984. So I guess I’ve had a thing for extracurricular math for a while.
I should check if you have anything on Clifford or geometric algebras. A little obsessed by those now.
Born 1966, did my math degree in Germany in 1991
Heh, I still have my high school math textbook from 1983 -- Algebra and Trigonometry 2, Dolciani, Wooten, Beckenbach,Sharron, Houghton-Mifflin 1971 ...
1959 and rediscovering my love of math thanks to you 🙂
I was born in 1960. I'm glad I'm old enough to remember what it was like before the internet, and compare it to what it is like afterwards. Those who have known nothing but the internet would do well to examine at the world before. Anyway, the infinite of maths available on the internet keeps me more than happy, as does this channel.
Born in 2003, absolutely love mathematics and I am hoping to become a theoretical physicist. Currently working through a book called tensors manifolds and forms, so your abstract algebra videos are a great way to wind down for me. thanks for the great content
I feel old now. Someone born in 2003 understanding tensors.
@@Pope_Balenciaga it’s in my course but In three years time when I study general relativity. I however love general mathematical structures as it often gives a deeper understanding to the framework of the model of physics we use. I also can’t wait for three years to study gr, so I’m studying it after I’ve finished that book haha
In 1983 I was in uni, studying number theory, data structures, and stuff; a little more than a decade earlier I figured out how to compute log[b](a) as a continued fraction.
My birth was before 1983, but I definitely am not old enough to have had a chance to solve this problem then!
This tournament happened 20 years before I was born.
1959 and can't get enough of Michael's videos.
I was one!!! Don’t recall much from that year of my life. I don’t remember being stressed at all!!!
Very nice calculations and usage of the blackboard with colours. I could follow all these steps, knowing the Vieta laws, binomic and trinomic formula etc. But I was wondering if the exercise from Austria has any deeper sense. Has the result a special meaning, or is it just an exercise for algebraic methods?
Born in 1954 and still loving Math and Sciences.
Born in 2007 and i love math
In 1983, I was studying calculus in college... 😁
13:22
Born in 1977 but still learning a lot from your channel.
Born 1963, Ph. D. in physics and re-gained my love of mathematics due to several math youtubers. One of them being Michael Penn-
Born in the 1950s, love math, in particular number theory, algebraic numbers, and abstract algebra.
Essentially writing the sum of cubes of the roots of a cubic polynomial in terms of the sum of its squares which may be obtained by expanding the square of the sum of the three roots.
I was 25 at the time!
(Actually, Michael, I don't think those from after "83 want to see things written with chalk on a board...)
11:00 Paused at that point. But after 3-5Minutes I found the solution by myself. It was easier than expected.
Has the link to suggest problems been taken down? I can't find it anymore.
It’s in the description at the very bottom.
-Stephanie
MP Editor
@@MichaelPennMath Thank you Stephanie.
a change of variable to x=y+1 followed by reducing the cubes to squares.. i dont know which could have been quicker
This for sure.
Nice video! It seems that there has been some effort lately to make prettier miniatures for the videos. That's great, but it has the annoying disadvantage for hiding the problem.
I'd prefer a miniature that gives me an idea of the content of the video before clicking
Vintage 1951 and I have always loved Mathematics.
Michael Penn stated at the end that x^3-6x^2+(41/5)x+41/5 = 0 has all complex solutions. Not true. A cubic equation has to have at least one real root.
This is a much nicer solution than my first thought, which was to use the cubic formula to get gnarly expressions for the roots in terms of a, plug into that equation relating the roots, and solve the resulting gnarly equation in a. If anyone likes pain they could try that. I'd be interested to know if there's a reason it couldn't work, just not enough to try it myself.
I used Vieta's formula to get a system of equations for the roots then used the cube of difference formula 3 times for the given equation. Using a;gebraic machinations I was able to get a equation for a and solve it. Got a = 123/15 in the end without finding every root individually
It's so satisfying getting such slightly ugly solution instead of boring 0 or 1.
Born before 83 and the eighties were great,if only i could go back to these years
Born in the sixties. Still love math. Love your channel.
Valuable content shared on a regular basis: what else could we ask for?
Born in 2004, so closer to now than 1983. But sorta crazy to think just barely
I see; so there are
*Section 1 - The System of Root - Coefficient Equations*
*Section 2 - The System of p(root) = 0 equations*
Let M be the matrix
[ 0 1 0 ]
[ 0 0 1 ]
[-a -a 6]
The characteristic polynomial of M is the polynomial that you have.
The whole point is to use the equation
Tr( (M-I)^3)=0.
You only need that
Tr(M)=6
Tr(M^2)=36-2a
and you will get an equation in a, without even touching alpha, beta, gamma.
You could do this more mechanically by just stating the condition for alpha, beta, and gamma in terms of elementary symmetric polynomials which are also equal to coefficients of the polynomial up to a sign. That's more work, though.
In 1983 I could count up to ten. It was a good start.
It seems like the demography of your audience is not what you thought!
I was born in 1973, which is cool because it is a prime number (in the last 100 years these years are also prime : 1931, 1933, 1949, 1951, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017)
Got BSc in 1982 (chemistry)
Was born ON 1983. July actually.
Hi Dr. Penn!
I very well could have participated in that Olympiad if I was Austrian (and had started school one year later)
was a traditional-age college student in 1983, majoring in math
love the penn fact
1985 here. Great video, Michael!
Only 2 of the roots are complex, and the real root is easy enough to extract via Cardano's Method.
I was at University doing Engineering in 1983 :)
Born in 1991, a nice palindromic number. I'm a certified theoretical physicist, but I'm enjoying pure mathematics more tan ever.
My birthday doesn't lie on the real line 😂🤣😂.
Born in 1976. In 1983 just finished my first year of elementary school 😁
Age in 1983 = X. 2X + 6 = age in 2023!
Thats my year !
I was born 25 Feburary 1983
Text-to-speech advert voice: Polynomials hate him!
I've been putting these questions in chatgpt and it solved this one instantly, I wonder if it's because someone else did or it's just really good at algebra. EDIT: now that I recall it doesn't use user inputs as training sets, it's just good at algebra, it knows vieta's formula and manipulates things like wolfram does. It's quite nice.
1983 was the year I learned the quadratic equation. So I guess that puts me on the early side…
Born in 2005, I will begin my computer science degree in the fall.
Born in 2003. Love your videos :)
pardon me ! whats the use of solving this ? what practical application does it solve ?
Born in 64. Didn't realise this was such an oldies channel until today.
Fixed the alpha squared....whew!
Watching from France and born in 65
I was born before 1983. Then again, I usually skip through the videos because I already know most of this stuff and have worked with very advanced mathematics. So, don't let me skew your demographics.
I was born exactly in 1983! very interesting comments to this one... 😂
I don't get the title
1983 i was -23 years old , yes with negative sign