I used to watch you a bunch in high school (2020-2022) when I was in pre calc (junior) and ap calc (senior). Although I could barely keep up with a lot of your videos because I hadn’t learned as much as I needed to understand it all, it was still super interesting and got me wanting to understand. I passed all of my classes with B-A’s, but quickly withdrew from math as a whole when I went into college mainly due to the difficultly with more new material and college taking more of my brain. I am studying psych and will get my bachelors in the winter of ‘24. However I have forgot most of calc as I didn’t need to take any more math classes either. However, I got Covid (again) and am near bed ridden till it goes away. And it sucks not being able to do anything productive. However, watching your videos makes me feel like I am doing something because I am trying to learn/re learn a lot of complex math. I mainly just wanted to say thankyou for making these videos. Even if I am in and out and not able (at the moment) to give your videos 100% effort for learning with trying the problems, they are still useful and interesting. I also appreciate you coming back to old videos with updates and more stuff for us to sink our teeth in. As well having these videos helped me a bit in my calc classes. Thankyou for making me feel a little bit smarter with each video and showing me the solutions and process to these seemingly impossible questions. Take care and continue being an absolute genius and wonderful person! ❤️ Happy holidays and a new years!
same but I loved maths when I was about 3-9 years old until my mom wanted me to "be better". Now I am form 1 and I rediscovered my interest towards maths because I standed out for myself.
Hey Ernest, I'm Ernest too, and I have always loved maths, but I've started to get seriously back into it over the last 10 years. I can't wait to retire (
@Chicken Leg No, it doesn't. He ain't a Japanese. You can find his biography and other credentials on the net. Check this one out: api-everipedia-org.cdn.ampproject.org/wp/s/api.everipedia.org/v2/wiki/amp/lang_en/blackpenredpen?usqp=mq331AQSKAFQApgBy46dyers4qcVsAEg
For those who want to know how to solve x^3 - x + 1 = 0, here is the method. First, note that (u + v)^3 = u^3 + 3vu^2 + 3uv^2 + v^3 = u^3 + v^3 + 3vu^2 + 3uv^2 = (u^3 + v^3) + 3uv(u + v). Therefore, let x = u + v. Hence x^3 = 3uvx + u^3 + v^3 = x - 1 implies 3uv = 1 and u^3 + v^3 = -1. 3uv = 1 implies v = 1/(3u), which implies u^3 + 1/(27u^3) = -1, which implies 27u^6 + 1 = -27u^3, which implies 27u^6 + 27u^3 + 1 = 0. Therefore, u^3 = [-27 + sqrt(621)]/54, or u^3 = [-27 - sqrt(621)]/54. From this, it can be shown rather trivially that v^3 is the conjugate of u^3. Therefore, without a loss of generality, u^3 = [-27 + sqrt(621)]/54 and v^3 = [-27 - sqrt(621)]/54, because addition is commutative. Therefore, u = -cbrt([sqrt(621) - 27]/54), and v = -cbrt([sqrt(621) + 27]/54), and as such, x = -[cbrt([sqrt(621) - 27]/54) + cbrt([sqrt(621) + 27]/54)] = -[cbrt([3·sqrt(69) - 27]/54) + cbrt([3·sqrt(69) + 27]/54)] = -[cbrt([sqrt(69) + 9]/18) + cbrt([sqrt(69) + 9]/18)]. This is the exact real answer, and you can show using some simple algebra that the result shown in the video actually simplifies to this. You don't need to actually know the cubic formula before hand. You only ever need to solve a system of equations to solve a cubic equation. In general, if you have a polynomial equation of the form y^3 = py + q, then by using the same method above, you can show that y = u + v, where 3uv = p, and u^3 + v^3 = q. Solving this system is always very easy as it reduces to solving a quadratic equation and taking the cube root of the solutions to that quadratic equation. Also, you can always convert a general polynomial equation x^3 + ax^2 + bx + c = 0 into the form above by letting x = y + a/3 and simplifying. Cubic equations are easier than they look. Quartic equations, on the other hand, are a genuine pain in the ass. Although they are always solvable, the method is significantly more tedious and annoying. EDIT: I made a dumb arithmetical mistake with my calculations, so I fixed it. EDIT 2: Split stuff into paragraphs, and fixed another dumb arithmetic mistake. See, this is why you don't do math after having pulled an all-nighter. *sigh*
Wow those expressions you added in red really makes a huge difference in factorization and it works so nicely. I like 6:35 the most. Def worth trying them out with hands for sure!
OMG! You actually did it twice! Soon, the whole world will know about that formula and the people behind them and the reason to teach the formula and the history instead of hiding it. Thank you so much!
Considering we ended up with an irrational value, a numeric solution would be as good. Any quintic with real coefficients has at least one real solution, so Newton-Raphson will give you a numeric solution. A quick inspection shows the expression changes sign between -1 and -2, so -1.5 would be a good starting point. I got -1.324717959 in four iterations. A numeric solution is not as "pure" as a formulaic solution, but the solution's the same and N-R is a _well-known_ _formula_ for solving these sort of problems.
Excelente lesson. That professor is perfect: he is smart and talented to teach, he sings, he plays multi color golf. What more? Very good!!! Greetings from Brasil. Thank you for your channel.
A quicker way to factor this problem is to use the zero property method at the beginning by adding and subtracting x^3 and then grouping (x^5+x^4+x^3) and (1-x^3).
I approximated the solution right away using Newton's method to be -1.32472 (in only 4 iterations), but of course I liked your way of solving it using factoring and then cubic formula. Interesting video!
My pre-view attack: x⁵ + x⁴ + 1 would be -cyclotomic- monic, with all coefficients = 1, if three more terms (x³ + x² + x) were added. So let's do this: *[Note: Correction - "cyclotomic" has a more specialized meaning, which coincides with the one I intended, only when the degree is p-1 for some prime, p.]* x⁵ + x⁴ + 1 = x⁵ + x⁴ + x³ + x² + x + 1 - x(x² + x + 1) . . . multiply by (x-1), which will introduce the extraneous root, x=1; we will remove this later. (x-1)(x⁵ + x⁴ + 1) = x⁶-1 - x(x³ - 1) = (x³-1)(x³ - x + 1) = (x-1)(x² + x + 1)(x³ - x + 1) So: x⁵ + x⁴ + 1 = (x² + x + 1)(x³ - x + 1) The zeros of the first (quadratic) factor are the 2 complex cube-roots of 1. So the desired real zero of the quintic is the real zero of the cubic factor. And the general cubic equation has a solution formula - see Angel Mendez-Rivera's comment; also check out the video by Mathologer where he shows how to develop the solution, while complaining that the cubic formula should be taught in algebra classes, alongside the quadratic solution formula. Of course, if you don't care to find the closed-form answer, you can always use Newton's Method, or some other iterative method to compute it to any desired precision. Fred
Great ideia! Would just like to point out that the polynomial Σx^i (i going up to n) is only cyclotomic iff n+1 is prime therefore that polynomial isn’t cyclotomic.
@@bernardopicao267 Yes. I used the wrong term for the right polynomial; will correct that. BTW, what is the right name for the general polynomial, of degree (integer) n ≥ 0, that is simply p(x;n) = (xⁿ⁺¹ - 1)/(x - 1) = ∑₀ⁿ xᵏ ? @Davis John: I challenge you to point out an error, other than the already-identified error of terminology brought up by Bernardo. In particular, my conclusion that the real zero of x⁵ + x⁴ + 1 is the (unique) real zero of x³ - x + 1. I believe it is you who are in error. Fred
It seems solution lie between -1 and -2 since x=-1 gives the value of equation is +1 and when x = -2 value is -15 then I can use bisection method -1.5 etc till I get close to zero
Since we have to use a calculator anyway.. Why not just use Newton raphson's method? It gives the answer in only 4 iterations provided initial root is taken as x=-1.
Well, I actually applied Newton's formula on the integral of the expression just to figure out where the minimum is located (the actual problem). After analyzing the derivative of the problem, (second derivative in Newton's method), I picked -1 as initial value. After 7 iterations of next = xi - f'(xi)/f''(xi) arrived to x approx. -1.325.
I used a trick my teacher explained: If the exponents of a polynomial are each congruent to a different number mod 3, then it is divisible by x^2+x+1. Example: x^5+x+x^0 or x^8+x^7+x^3. It works for other number too. x^16 + x^11+x^6+x is divisible by x^3+x^2+x+1. This way I could factor the equation in a cubic and a quadratic equation. Unfortunately couldn't solve the cubic part, but at least I could find two solutions
@@ottoaberg5942 mod is short for modulus, the integer remainder of integer division. So 5 mod 3 = 2, 4 mod 3 = 1, and 0 mod 3 = 0. Since the answers differ the OP claims it is divisible by x^2+x+1.
@@ottoaberg5942 That is the assertion made by OP, and a quick check with Mathematica shows it seems to work. But I've not seen the proof for this rule myself...
in high school we were taught to factor any polynomial by dividing it by (x-b) where b is a factor of the 0-th term; that is 1 in your example. so we divided once by (x-1) and again by (x+1).
What a nice guy! I converted the polynomial to a "depressed quintic" (sigh) y^5+y+1=0 with y=1/x. I guessed there would be roots y and y* with |y|=1. y=-1/2 +/- i sqrt(3)/2 turns out to work. Multiplying the terms for these gives the quadratic factor leaving the depressed cubic (double sigh). After that I Googled
In the beginning, I tried to write it as (x^4+1)(x+1), then I tried something with (x^3+1) and then think to something which make some terms simplifies and in the end find the answer. Nice, but Easy one.
Another interesting way to solve this equation is to use the symmetry of its solutions. Let z a solution of the equation Then to have z⁵+z⁴+1 z⁵ and z⁴ can be complex conjugate each other. Let ⃗z the conjugate of z. Then ⃗z⁴ will be conjugate of z⁴ then equal to z⁵ Then we have this awesome equation z⁵=⃗z⁴. Using exponential notation for complex numbers r⁵exp(5θ)=r⁴exp(-4θ) for r≠0 then r=1 to satisfy the equation and 5θ=-4θ+2πk we found that θ=2π/3 is a solution. then z=exp(±2π/3) are two solutions of the equation. Then since z are two cubic roots of the unit they satisfy the equation. z²+z+1=0 then x²+x+1 divide x⁵+x⁴+1.
I solved this by recognising almost immediately (by inspection since if we set x^3 = 1, we get x^2 + x + 1 =0 which also returns the conjugate complex roots of one) that that the two complex cube roots of one solved this. Which means linear factors of (x-omega) and (x-omega^2). Then by serial synthetic division and using the identities related to the complex cube roots of one, I got x^3 - x +1 =0. I solved this using the cubic solution I derived from first principles (recognising the reduced cubic form and putting x = z + 1/(3z) followed by z^3 = m and solving the quadratic). It was after this last step (when I'd already got the correct solution) that I watched your video hoping for more insight or a clever trick in how you solved that final cubic, but alas - you just regurgitated the cubic formula.
It's a shame that I don't have anything to comment about before coming here, but I can't miss this golden opportunity to be this early lol.. I just want to say that I really like your videos and even though I still haven't officially learned calculus or trigonometry in school I watch every single one of your videos! :D
@@einsteingonzalez4336 That's what I like about these videos. Thanks to it, I think it will be easy learning complex numbers, since I already know the basics of what it is and how to use it etc, plus I see it everywhere and it becomes normal at this point! Same with trigonometry. I started to understand the small stuff that no one pays attention to, like the angles of pi related to the circle, euler's formula, comparing numbers, tricks to solving equations, and I can state many more.
Numberphile has a stimulating video on the plastic ratio, and the Wikipedia page on the plastic number (the same thing under a different name) is pretty good. Also, typing "plastic number" into Wolfram Alpha and then clicking on "More information" provides the start of an interesting mathematical tour.
The solution is the negative of the plastic constant p, which solves the equation x^3 - x - 1 = 0 (equivalent to (x-1)(x)(x+1)=1 ), also solves the equation x^5 - x^4 - 1 = 0 (this can be obtained from bprp's equation by substituting u = -x) and is the limiting ratio of the Padovan and Perrin sequences which are analogous to the Fibonacci sequence and follow the rule F(n) = F(n-2) + F(n-3) = F(n-1) + F(n-5).
tha's what I did too. It was much faster. Unfortunately I got stuck because I did not know the cubic formula. I was hoping to see a solution that didn't require it. oh well.
I just tested a bunch of numbers with two columns. One for x and one for f(x), and I got pretty close. I got to 1.32 which would be 0.02, which is quite close. I'm sure you'd get the answer eventually that way if you had enough patience.
x⁵ + x⁴ + 1 = 0 ⇒ x³(x² + x + 1) - x³ + 1 = 0 ⇒ x³(x² + x + 1) - (x³ - 1) = 0 ⇒ x³(x² + x + 1) - (x -1)(x² + x + 1) = 0 (because a³ - b³ = (a - b)(a² + ab + b²)); then we have (x² + x + 1)(x³ - x + 1) = 0 ⇒ x³ - x + 1 = 0, because x² + x + 1 = 0 does not have any real solution ( Δ = - 3 < 0) to be continued ... P.S: we need to note that if f(x) = x⁵ + x⁴ + 1 , then f(x) = 0 has only one real solution, and this solution is negative, between (-2) and (-1)); ... ... and this in using lim f(x) when x → - ∞; lim f(x) when x → + ∞; and f'(x) = 5x⁴ + 4x³ = x³(5x + 4) = x(5x + 4).x² x³ - x + 1 = 0 is easy to solve with Cardan's formula.
You are very good and lucky to factorised the PENTIC=cubic*quadratic. Using remainder theorem f=pq+r .Within 5 seconds testing 0 r+ , -1 r+ , -2 r-. immediately the only Real solution between x=-1 and x=-2 .Since x=a+bi and x=c+di . Thus the best way is to use NewtonRaphson Method . follow by using scientisfic-Calculator or ComputerExcel .You can get answer-12digits or more Accuracy/Precision.
I don't think you can go much further with that except compluter. In this process ultimately you have to solve this to find the intersection analytically
x⁵ + x⁴ + 1 = 0 ⇒ x³(x² + x + 1) - x³ + 1 = 0 ⇒ x³(x² + x + 1) - (x³ - 1) = 0 ⇒ x³(x² + x + 1) - (x -1)(x² + x + 1) = 0 (because a³ - b³ = (a - b)(a² + ab + b²)); then we have (x² + x + 1)(x³ - x + 1) = 0 ⇒ x³ - x + 1 = 0, because x² + x + 1 = 0 does not have any real solution ( Δ = - 3 < 0) to be continued ... P.S: we need to note that if f(x) = x⁵ + x⁴ + 1 , then f(x) = 0 has only one real solution, and this solution is negative, between (-2) and (-1)); ... ... and this in using lim f(x) when x → - ∞; lim f(x) when x → + ∞; and f'(x) = 5x⁴ + 4x³ = x³(5x + 4) = x(5x + 4).x² x³ - x + 1 = 0 is easy to solve with Cardan's formula.
Me, being an engineer: Ok... google, what was the formula for Newton's method again? ...calculator goes brrr... Hmmm, something around -1.325. Lets put -1.5, it will do its job.
Hahaha i did the exact same thing took a couple seconds. Ahh it's around -1 , close enough. If we'd worked together we would have averaged it and nailed it.
Although there is not an analytical expression for the solutions of quintic equations, there are conditions on the existence of closed expressions for certain coefficients. Those I think were developed by Abel, but I'm not quite sure.
- 1,324718 My procédure was different, based on complex numbers. I differentiated once, twice, etc. and observed there was only one real root. So there are complex roots. Then I tried the assumption that two of the complex ones were on the unit circle, i.e. x=exp(+/- i thêta). In order for exp(i 5theta) +exp(i 4theta) to equ)al -1, aka exp(i π), we need sin(4theta) = - sin(5theta) and cos(4theta) + cos(5theta) = -1 Implying cos(4theta) = cos(5theta) = - 1/2 Which (a graphic is useful here) gives theta = +/- (2/3)π = +/- 120° or x = (1/2) +/- i (V3/2) or, combined, x²+x+1=0. The assumption proved correct. Dividing the original equiation by this factor leads to the cubic solving following your demonstration. Calling the two cube roots a and b, we have x0 = a+b and x1,x2 = (a+b)/2 +/- i |a-b|/2
Note that x^3=1 has three solutions: 1, w, w^2, we have w^5+w^4+1=w^2+w+1=(w^3-1)/(w-1)=0 and w^10+w^8+1=w+w^2+1=(w^3-1)/(w-1)=0. Therefore w and w^2 are the solutions of the function x^5+x^4+1=0. Thus the polynomial x^5+x^4+1 has factor x^2+x+1. One could simply conclude that x^5+x^4+1=(x^2+x+1)(x^3-x+1)
I have a simpler way, you can derive the equation and find the extrim point to be x=0,x=-0.8 , by ploting it you see that at x=-0.8 you get y=0 ( a solution) at x=0 you get y=1, it is also noticible that the derivative of the function is 0 at this points and positive elsewere, that meens that this is a monotoic function and the only solution is at x=-0.8
I would have cranked the equation into an online polynomial solver, like Wolfram, it would give in two seconds the five solutions as stars in a nice complex plane. By the way, do you know that the family of polynomials equations: x-1=0, x^2+x-1=0, x^3+x^2+x-1=0, .... are related with the Fibonacci numbers, and of course with the Golden Ratio??
@@jimgraham6722 :: The number of iterations for convergence for Newton Rapson depends on the seed value you give in as Xo, and the precision you need. And also not to fall in a zone where the function has a valley, where the iterations will go up and down, oscillating.
@@powerdriller4124 Yes indeed, NM has some limitations, but to me it is fun to watch it resolve some otherwise difficult problems using a cheap programmable calculator.
One solution is cube root of unity that is -1+√3i/2,-1-√3i/2 try putting them by x^3=-1 According to complex comcepts w,w^2 are roots of unity and w^3=1 and 1+w+w^2=0 Therefore the equation x^5+x^4+1=0 becomes x^2.x°3+x.x^3+1=0 now x^3=1 for x=w therefore equation becomes 1+w+w^2 HP
Equation with only 1, -1, and 0 : try complex roots of 1, say a , and 1/a must be solution so we get a new polynomial equation: 1+a+a^5=0. the difference yield the equation : a^4-a=0, a(a-1)(a^2+a+1)=0 : and try factoring the original equation by a^2+a+1.
This is really genius. All is derived in a logical and understandable manner. I tried to figure out the polynomial multipöication but it got to complicated. Still I had the idea that maybe x^2 + 1 or x^2 + x + 1 could be the quadratic factor, since both of these end with 1 and have no real solution. But the grouping scheme was great and much easier!
Given that there is only one real non-repeated solution, this means it has to come from the cubic factor and not from the quadratic. This means that the cubic factor is a case where Cardano's formula works directly, because the cubic discriminant will be positive for the single non-repeated root. The quadratic factor could give us two real roots, or a real/repeated root, but not just one real/distinct root. If there were only one real root and it were repeated, it could come from the quadratic factor. This isn't possible with an odd degree polynomial, since a repeated root means there's a turning point at the root. It has to turn back around and cross the x-axis again. So any odd-degree polynomial with a real repeated root, will have another distinct root as well.
We can also use the concept of primitive root of unity. Suppose that w^3 = 1; w is the primitive cube root of unity =/ 1. So w^5 + w^4 + 1 = w^3 + w^2 + 1 => (x^2 + x + 1) is a factor.
whenever u have higher degree polynomials just check ones iota negative iota cube root of unity and square of cube root of unity and negatives of it..... i just checked cube root of unity and square of it was the solution of this equation. So i just divided the polynomial by x^2 + x + 1 and it will be divisible because zeroes of this polynomial are cube root of unity and its squared and the actuall polynomial zeroes were also the same, so after dividing we just have a cubic equation to solve.......
Here is another trick to find the factorization of the polynomial. Let a be a primitive third root of unity. Hence a^3=1 and a^2+a+1=0. Substituting x=a in the given equation we get a^5+a^4+1=a^2+a+1=0. Hence both the primitive third roots must be roots of the given polynomial also. Hence x^2+x+1 must divide the given polynomial. The other factor is obtained easily by long division.
Note that both terms inside the cube roots are negative so if you are trying to calculate this numerically it might be useful to express them as the negative of the cube root of their absolute value otherwise you might end up picking one of the complex cube roots _i.e._ -((1/2 - sqrt(23/108))^(1/3)) - (1/2 + sqrt(23/108))^(1/3) ~ 1.32472
I could see that "w and w^2" are the solutions of the original equation, coz w^2 + w + 1 = 0 and w^3 = 1. So I got x^2 + x + 1 as a factor directly. Works in this particular case only though 😅
Divide through by x^4, get the equation(1) x = -1 -1/x^4; Guess that the root must be negative, because x^4 is always positive. observe that x^5 - x^4 = -1. Guess x = -1.2; now iterate (1) starting with x= -1.2 and the answer will eventually converge to -1.324717957.....You can improve the rate of convergence by averaging the input and output numbers each time..
That solution, -1.3247, is the negative of the famous plastic ratio given by x^5 - x^4 - 1 = 0, the ratio constant of the Padovan sequence with the recurrence relation A(n) = A(n-2) + A(n-3). What famous ratio solves this quintic: x^5 - x^3 - x^2 - x^1 - x^1 - x^0 = 0? (Note the sequence of exponents).
if w is the complex cube root of unity ... it is obvious both w and w^2 are roots of x^5+x^4+1 ..so it is obvious that (x-w)(x-w^2)=x^2+x+1 is a factor.. A similar logic applies when you are asked to determine if weird numbers like 1280000401are prime or not. The trick is to see if the number can be written as a polynomial x^(2 mod 3) + x^( 1 mod 3) + 1 ...then by the above logic it has to be divisible by x^2+x+1 ... for example the number above is 20^7+20^2+1 .. hence divisible by 421=20^2+20+1.
0:45 I don't have a formula for the general case, but at least I figured out that by proceeding in the same way as in the video, one can factor x^5 + ax^4 + a^5 into (x^2 + ax + a^2) (x^3 - a^2 x + a^3).
To solve this équation it possible to write two fonctions y = x^4 and Other fonction y = - x^5 - 1 and built their curves and the solution is the point sur of their intersection .
Try this extreme quintic equation: th-cam.com/video/GoGsVLnf8Rk/w-d-xo.html
I used to watch you a bunch in high school (2020-2022) when I was in pre calc (junior) and ap calc (senior). Although I could barely keep up with a lot of your videos because I hadn’t learned as much as I needed to understand it all, it was still super interesting and got me wanting to understand. I passed all of my classes with B-A’s, but quickly withdrew from math as a whole when I went into college mainly due to the difficultly with more new material and college taking more of my brain. I am studying psych and will get my bachelors in the winter of ‘24. However I have forgot most of calc as I didn’t need to take any more math classes either.
However, I got Covid (again) and am near bed ridden till it goes away. And it sucks not being able to do anything productive. However, watching your videos makes me feel like I am doing something because I am trying to learn/re learn a lot of complex math. I mainly just wanted to say thankyou for making these videos. Even if I am in and out and not able (at the moment) to give your videos 100% effort for learning with trying the problems, they are still useful and interesting. I also appreciate you coming back to old videos with updates and more stuff for us to sink our teeth in. As well having these videos helped me a bit in my calc classes. Thankyou for making me feel a little bit smarter with each video and showing me the solutions and process to these seemingly impossible questions. Take care and continue being an absolute genius and wonderful person! ❤️ Happy holidays and a new years!
I am 65 years old. When I was young, I loved math. Thank you, I just rediscovered. :-) a Slovak pensioner.
Odkiaľ?
same but I loved maths when I was about 3-9 years old until my mom wanted me to "be better". Now I am form 1 and I rediscovered my interest towards maths because I standed out for myself.
comment edited due to grammar issues
I am sexy five and still love math, Ans still think world revolve around math ...
Hey Ernest, I'm Ernest too, and I have always loved maths, but I've started to get seriously back into it over the last 10 years. I can't wait to retire (
Is no one going to talk about those marker skills?? Man is blowing my mind with those swaps
Absolutely filthy marker control 😂😂😂
That's what the channel name says. "Blackpenredpen"
He's going to be a legend!
I can't understand anything. I'm just watching how he changes marker.
@Chicken Leg No, it doesn't. He ain't a Japanese. You can find his biography and other credentials on the net.
Check this one out: api-everipedia-org.cdn.ampproject.org/wp/s/api.everipedia.org/v2/wiki/amp/lang_en/blackpenredpen?usqp=mq331AQSKAFQApgBy46dyers4qcVsAEg
He's really good at maths, but I'm equally impressed with his writing too.
Thanks.
Yaa writing with 2 pens simultaneously is really a great skill..
most real math teachers usually have a good and decent handwriting.
@@XBGamerX20And what about the complex math teachers?
@@antoniusnies-komponistpian2172 & what about the rational math teachers?
For those who want to know how to solve x^3 - x + 1 = 0, here is the method.
First, note that (u + v)^3 = u^3 + 3vu^2 + 3uv^2 + v^3 = u^3 + v^3 + 3vu^2 + 3uv^2 = (u^3 + v^3) + 3uv(u + v). Therefore, let x = u + v. Hence x^3 = 3uvx + u^3 + v^3 = x - 1 implies 3uv = 1 and u^3 + v^3 = -1. 3uv = 1 implies v = 1/(3u), which implies u^3 + 1/(27u^3) = -1, which implies 27u^6 + 1 = -27u^3, which implies 27u^6 + 27u^3 + 1 = 0.
Therefore, u^3 = [-27 + sqrt(621)]/54, or u^3 = [-27 - sqrt(621)]/54. From this, it can be shown rather trivially that v^3 is the conjugate of u^3. Therefore, without a loss of generality, u^3 = [-27 + sqrt(621)]/54 and v^3 = [-27 - sqrt(621)]/54, because addition is commutative.
Therefore, u = -cbrt([sqrt(621) - 27]/54), and v = -cbrt([sqrt(621) + 27]/54), and as such, x = -[cbrt([sqrt(621) - 27]/54) + cbrt([sqrt(621) + 27]/54)] = -[cbrt([3·sqrt(69) - 27]/54) + cbrt([3·sqrt(69) + 27]/54)] = -[cbrt([sqrt(69) + 9]/18) + cbrt([sqrt(69) + 9]/18)]. This is the exact real answer, and you can show using some simple algebra that the result shown in the video actually simplifies to this.
You don't need to actually know the cubic formula before hand. You only ever need to solve a system of equations to solve a cubic equation. In general, if you have a polynomial equation of the form y^3 = py + q, then by using the same method above, you can show that y = u + v, where 3uv = p, and u^3 + v^3 = q. Solving this system is always very easy as it reduces to solving a quadratic equation and taking the cube root of the solutions to that quadratic equation.
Also, you can always convert a general polynomial equation x^3 + ax^2 + bx + c = 0 into the form above by letting x = y + a/3 and simplifying. Cubic equations are easier than they look.
Quartic equations, on the other hand, are a genuine pain in the ass. Although they are always solvable, the method is significantly more tedious and annoying.
EDIT: I made a dumb arithmetical mistake with my calculations, so I fixed it.
EDIT 2: Split stuff into paragraphs, and fixed another dumb arithmetic mistake. See, this is why you don't do math after having pulled an all-nighter. *sigh*
So amazing!!
blackpenredpen Thank you!
@@blackpenredpen I never came across such formula so thnks
That effort to write this should be appreciated
Nice! If you want to convert a general cubic to the depressed form, I think you have to do the b/3a trick
Anybody else appreciate the fact that this man held a pokeball for the duration of this video
It's a mic :)
@@dillonroller a very cool mic.\
It's the only reason I watched to the end.
Great! Thank's!
Yes it’s so nice
Wow those expressions you added in red really makes a huge difference in factorization and it works so nicely. I like 6:35 the most. Def worth trying them out with hands for sure!
So true. That's what I felt
MathFlix
Thanks!!
0.1123671011141920252829323742. True fibonacci sequence.
MarkusDarkess Not true Fibonacci sequence. More like Tribonacci sequence, but that wouldn't be correct either.
@@blackpenredpen take a look at jee exam india
OMG! You actually did it twice!
Soon, the whole world will know about that formula and the people behind them
and the reason to teach the formula and the history instead of hiding it.
Thank you so much!
U mean on my Chinese channel too?
@@blackpenredpen Yep, that too. : )
@@blackpenredpen Wait, remember the video when you proved that sin(10°) is irrational?
th-cam.com/video/cN0vDlWjFiY/w-d-xo.html
Uea
@@blackpenredpen Yep. Use Cardano's formula to find sin(10°).
After all, it's the solution to 3x - 4x^3 = 1/2.
"There is no known formula for this. Anyways, please pause the video and try to solve."
You give me too much credit, sir
Considering we ended up with an irrational value, a numeric solution would be as good. Any quintic with real coefficients has at least one real solution, so Newton-Raphson will give you a numeric solution. A quick inspection shows the expression changes sign between -1 and -2, so -1.5 would be a good starting point. I got -1.324717959 in four iterations. A numeric solution is not as "pure" as a formulaic solution, but the solution's the same and N-R is a _well-known_ _formula_ for solving these sort of problems.
Excelente lesson. That professor is perfect: he is smart and talented to teach, he sings, he plays multi color golf. What more? Very good!!! Greetings from Brasil. Thank you for your channel.
A quicker way to factor this problem is to use the zero property method at the beginning by adding and subtracting x^3 and then grouping (x^5+x^4+x^3) and (1-x^3).
But how?
give this man a bigger white board lmao
No hace falta... necesita mejor camarografo🥴
Chopin I'm your biggest fan
🤣🤣🤣🤣
🤣🤣🤣🤣🤣after that he will solve long long eqn in a single step..
And different shades of red pen
I approximated the solution right away using Newton's method to be -1.32472 (in only 4 iterations), but of course I liked your way of solving it using factoring and then cubic formula. Interesting video!
Can we just take a moment to appreciate how beautiful his squared/cubed roots are
Did anyone noticed his speed of changing the marker so fast and perfect..👀....Amazing man 🔥🔥😊
My pre-view attack:
x⁵ + x⁴ + 1 would be -cyclotomic- monic, with all coefficients = 1, if three more terms (x³ + x² + x) were added. So let's do this:
*[Note: Correction - "cyclotomic" has a more specialized meaning, which coincides with the one I intended, only when the degree is p-1 for some prime, p.]*
x⁵ + x⁴ + 1 = x⁵ + x⁴ + x³ + x² + x + 1 - x(x² + x + 1) . . . multiply by (x-1), which will introduce the extraneous root, x=1; we will remove this later.
(x-1)(x⁵ + x⁴ + 1) = x⁶-1 - x(x³ - 1) = (x³-1)(x³ - x + 1) = (x-1)(x² + x + 1)(x³ - x + 1)
So:
x⁵ + x⁴ + 1 = (x² + x + 1)(x³ - x + 1)
The zeros of the first (quadratic) factor are the 2 complex cube-roots of 1. So the desired real zero of the quintic is the real zero of the cubic factor.
And the general cubic equation has a solution formula - see Angel Mendez-Rivera's comment; also check out the video by Mathologer where he shows how to develop the solution, while complaining that the cubic formula should be taught in algebra classes, alongside the quadratic solution formula.
Of course, if you don't care to find the closed-form answer, you can always use Newton's Method, or some other iterative method to compute it to any desired precision.
Fred
Enclosing the missing root is beautiful, like inserting the final stone into a mosaic. Thanks for sharing!
Great ideia! Would just like to point out that the polynomial Σx^i (i going up to n) is only cyclotomic iff n+1 is prime therefore that polynomial isn’t cyclotomic.
Sir I am sorry to inform you that your “pre-view attack” is wrong. Nice try, however.
Show off.
@@bernardopicao267 Yes. I used the wrong term for the right polynomial; will correct that.
BTW, what is the right name for the general polynomial, of degree (integer) n ≥ 0, that is simply p(x;n) = (xⁿ⁺¹ - 1)/(x - 1) = ∑₀ⁿ xᵏ ?
@Davis John: I challenge you to point out an error, other than the already-identified error of terminology brought up by Bernardo.
In particular, my conclusion that the real zero of x⁵ + x⁴ + 1 is the (unique) real zero of x³ - x + 1.
I believe it is you who are in error.
Fred
He is like the Bob Ross of Math! How do you make math so much fun?
It seems solution lie between -1 and -2 since x=-1 gives the value of equation is +1 and when x = -2 value is -15 then I can use bisection method -1.5 etc till I get close to zero
Since we have to use a calculator anyway.. Why not just use Newton raphson's method? It gives the answer in only 4 iterations provided initial root is taken as x=-1.
Me: By intermediate value theorem, the root must lie between -2 and -1. QED
Come on. A good engineer would at least limit that range to (-1.5,-1.25) :)
Me: by definition, the real root must be a real number. QED.
How about solving this numerically? NEWTON'S METHOD... lol
Lol by Extreme Value Theorem, there is at least 1 relative min and relative max :)
The first think I managed to do is to estimate the root and to prove that there is only one real root
Well, I actually applied Newton's formula on the integral of the expression just to figure out where the minimum is located (the actual problem). After analyzing the derivative of the problem, (second derivative in Newton's method), I picked -1 as initial value. After 7 iterations of next = xi - f'(xi)/f''(xi) arrived to x approx. -1.325.
Wow. Are you Spanish?. No wonder Spanish people are so smart with good jobs. Great
"yeah I did quadratic equations in high school"
they said
"it looks pretty similar"
they said
"it can't be too difficult"
they said
They said...
th-cam.com/video/kow8ijXyVMQ/w-d-xo.html
Tartaglia... That's a name I haven't heard in a long long time
Seriously, his name deserves to be mentioned more.
Gives me Godfather feels :)
@@guitarbonanzabychiragkar4249 I've now got half a mind to do a video about Cardano's formula in a very bad godfather voice
@@guitarbonanzabychiragkar4249
Yes, me too. (That was Tattaglia though, of course.)
I wonder if anybody else knows what it means...
I used a trick my teacher explained: If the exponents of a polynomial are each congruent to a different number mod 3, then it is divisible by x^2+x+1. Example: x^5+x+x^0 or x^8+x^7+x^3. It works for other number too. x^16 + x^11+x^6+x is divisible by x^3+x^2+x+1. This way I could factor the equation in a cubic and a quadratic equation. Unfortunately couldn't solve the cubic part, but at least I could find two solutions
Nome de Brasileiro kkk
What does "congruent to a different number mod 3" mean? My english is lacking
@@ottoaberg5942 mod is short for modulus, the integer remainder of integer division. So 5 mod 3 = 2, 4 mod 3 = 1, and 0 mod 3 = 0. Since the answers differ the OP claims it is divisible by x^2+x+1.
@@ronaldjensen2948 i see. so a polynomial that includes the exponents 11 and 5 would not follow this rule since the remainder is 2 for both of them?
@@ottoaberg5942 That is the assertion made by OP, and a quick check with Mathematica shows it seems to work. But I've not seen the proof for this rule myself...
Truly mesmerized by the technique.
That moment when you have to leave in 5 minutes but BPRP just released a new 10 minute video
Play at double speed
x2 speed
I watch all his vids in x2.
It's very accurately states all the points and their margins. Great !
i love it when u use different colours to write the different parts, it really makes it much more clear!
in high school we were taught to factor any polynomial by dividing it by (x-b) where b is a factor of the 0-th term; that is 1 in your example. so we divided once by (x-1) and again by (x+1).
-1 is pretty close by guessing. Then do 5 Newton iterations and wala. But your method is pretty good as well.
What a nice guy! I converted the polynomial to a "depressed quintic" (sigh) y^5+y+1=0 with y=1/x. I guessed there would be roots y and y* with |y|=1. y=-1/2 +/- i sqrt(3)/2 turns out to work. Multiplying the terms for these gives the quadratic factor leaving the depressed cubic (double sigh). After that I Googled
In the beginning, I tried to write it as (x^4+1)(x+1), then I tried something with (x^3+1) and then think to something which make some terms simplifies and in the end find the answer. Nice, but Easy one.
Another interesting way to solve this equation is to use the symmetry of its solutions.
Let z a solution of the equation
Then to have z⁵+z⁴+1
z⁵ and z⁴ can be complex conjugate each other.
Let ⃗z the conjugate of z. Then ⃗z⁴ will be conjugate of z⁴ then equal to z⁵
Then we have this awesome equation
z⁵=⃗z⁴.
Using exponential notation for complex numbers
r⁵exp(5θ)=r⁴exp(-4θ)
for r≠0 then r=1 to satisfy the equation and
5θ=-4θ+2πk
we found that θ=2π/3 is a solution.
then z=exp(±2π/3) are two solutions of the equation.
Then since z are two cubic roots of the unit they satisfy the equation.
z²+z+1=0
then x²+x+1 divide x⁵+x⁴+1.
like the fact he always looks happy
I solved this by recognising almost immediately (by inspection since if we set x^3 = 1, we get x^2 + x + 1 =0 which also returns the conjugate complex roots of one) that that the two complex cube roots of one solved this. Which means linear factors of (x-omega) and (x-omega^2). Then by serial synthetic division and using the identities related to the complex cube roots of one, I got x^3 - x +1 =0. I solved this using the cubic solution I derived from first principles (recognising the reduced cubic form and putting x = z + 1/(3z) followed by z^3 = m and solving the quadratic). It was after this last step (when I'd already got the correct solution) that I watched your video hoping for more insight or a clever trick in how you solved that final cubic, but alas - you just regurgitated the cubic formula.
What a cool quarantine quintic!
Divide equation by x^2: x^3+x^2+1/x^2, which implies x^3+(x+1/x)^2-2=x^3-1+(x+1/x)^2-1 so factoring this and dividing by x^2-x+1 we get desired cubic
You know that you are a math enthusiast when you casually present a quintic equation.
No words to say , u are great sir.
We solve as math teachers cubics as well
You might look at some examples
th-cam.com/video/ircZ_M1m3I8/w-d-xo.html
It's a shame that I don't have anything to comment about before coming here, but I can't miss this golden opportunity to be this early lol..
I just want to say that I really like your videos and even though I still haven't officially learned calculus or trigonometry in school I watch every single one of your videos! :D
Yes. They're explained so easily that ANYONE can understand it.
@@einsteingonzalez4336 That's what I like about these videos.
Thanks to it, I think it will be easy learning complex numbers, since I already know the basics of what it is and how to use it etc, plus I see it everywhere and it becomes normal at this point!
Same with trigonometry.
I started to understand the small stuff that no one pays attention to, like the angles of pi related to the circle, euler's formula, comparing numbers, tricks to solving equations, and I can state many more.
Numberphile has a stimulating video on the plastic ratio, and the Wikipedia page on the plastic number (the same thing under a different name) is pretty good. Also, typing "plastic number" into Wolfram Alpha and then clicking on "More information" provides the start of an interesting mathematical tour.
Thanks for the recommendations!!
You know what is happiness? It's to know I never have to solve such things in my life. Pure joy. ^__^
The solution is the negative of the plastic constant p, which solves the equation x^3 - x - 1 = 0 (equivalent to (x-1)(x)(x+1)=1 ), also solves the equation x^5 - x^4 - 1 = 0 (this can be obtained from bprp's equation by substituting u = -x) and is the limiting ratio of the Padovan and Perrin sequences which are analogous to the Fibonacci sequence and follow the rule F(n) = F(n-2) + F(n-3) = F(n-1) + F(n-5).
Adding and substacting x^2 also works.
tha's what I did too. It was much faster. Unfortunately I got stuck because I did not know the cubic formula. I was hoping to see a solution that didn't require it. oh well.
I have a nack for picking out successful TH-cam channels. The common factor in these channels is passion.
I just tested a bunch of numbers with two columns. One for x and one for f(x), and I got pretty close. I got to 1.32 which would be 0.02, which is quite close. I'm sure you'd get the answer eventually that way if you had enough patience.
I'm in highschool. I'm so passionate about maths. I hope I'm as good as you are when I'm your age. Absolutely surreal
Good luck! I hope you studies go great!!
I was like, well, we know -2 < x < -1, and then I gave up and watched the video
Same lol
I found the approx. solution by the binary search.
Kinda looks like a cheating, but works too.
>>> y=lambda x: x**5+x**4+1
>>> y(-2)
-15
>>> y(-1.9)
-10.728889999999994
>>> y(-1.7)
-4.84647
>>> y(-1.3)
0.14317000000000002
>>> y(-1.4)
-0.5366399999999989
>>> y(-1.35)
-0.16252718750000072
...
>>> y(-1.325)
-0.0017212207031245264
...
>>> y(-1.3247)
0.00010951904337330731
@@victorshilin9360 ya that is cheating lol
Victor Shilin This is how my brain works
x⁵ + x⁴ + 1 = 0 ⇒ x³(x² + x + 1) - x³ + 1 = 0 ⇒ x³(x² + x + 1) - (x³ - 1) = 0 ⇒ x³(x² + x + 1) - (x -1)(x² + x + 1) = 0 (because a³ - b³ = (a - b)(a² + ab + b²));
then we have (x² + x + 1)(x³ - x + 1) = 0 ⇒ x³ - x + 1 = 0, because x² + x + 1 = 0 does not have any real solution ( Δ = - 3 < 0)
to be continued ...
P.S: we need to note that if f(x) = x⁵ + x⁴ + 1 , then f(x) = 0 has only one real solution, and this solution is negative, between (-2) and (-1)); ...
... and this in using lim f(x) when x → - ∞; lim f(x) when x → + ∞; and f'(x) = 5x⁴ + 4x³ = x³(5x + 4) = x(5x + 4).x²
x³ - x + 1 = 0 is easy to solve with Cardan's formula.
Your solution is very nice man
honestly I am just enjoying how he writes on a white board at this point
You are very good and lucky to factorised the PENTIC=cubic*quadratic. Using remainder theorem f=pq+r .Within 5 seconds testing 0 r+ , -1 r+ , -2 r-. immediately the only Real solution between x=-1 and x=-2 .Since x=a+bi and x=c+di . Thus the best way is to use NewtonRaphson Method . follow by using scientisfic-Calculator or ComputerExcel .You can get answer-12digits or more Accuracy/Precision.
Maybe, we could get away with some graphing. Take x⁵+x⁴ on lhs and -1 on rhs and we could surely make a good enough approximation
Nice approach brother
I don't think you can go much further with that except compluter. In this process ultimately you have to solve this to find the intersection analytically
Okay, but what if we don't want an approximation?
@@angelmendez-rivera351 nothing is accurate
x⁵ + x⁴ + 1 = 0 ⇒ x³(x² + x + 1) - x³ + 1 = 0 ⇒ x³(x² + x + 1) - (x³ - 1) = 0 ⇒ x³(x² + x + 1) - (x -1)(x² + x + 1) = 0 (because a³ - b³ = (a - b)(a² + ab + b²));
then we have (x² + x + 1)(x³ - x + 1) = 0 ⇒ x³ - x + 1 = 0, because x² + x + 1 = 0 does not have any real solution ( Δ = - 3 < 0)
to be continued ...
P.S: we need to note that if f(x) = x⁵ + x⁴ + 1 , then f(x) = 0 has only one real solution, and this solution is negative, between (-2) and (-1)); ...
... and this in using lim f(x) when x → - ∞; lim f(x) when x → + ∞; and f'(x) = 5x⁴ + 4x³ = x³(5x + 4) = x(5x + 4).x²
x³ - x + 1 = 0 is easy to solve with Cardan's formula.
We can also do using Descarts rule of signs to know how many real roots
0:41 you can substitute x=1/y to get an equation which only has the linear and constant terms, which (in some cases) should be solvable
How would this work? Bello trovarsi tra italiani nei commenti 👍
If we subtitute x by 1/y:
Y^(-5)+y^(-4)+1=0
«=»1+y+y^5=0
I don't think it s easier
@@rainbow-cl4rk its not easier, but there are formulas for some cases of quintics in the form x⁵+ax+b=0
@@matteodamiano7598 ah ok, tanks, i didn't know that
@@matteodamiano7598 However the conditions for the formulas to work are... well, let's call them challenging to compute.
I think you should use Jacobi theta funcion , hypergeometric functions or Mellin integral
General quintic is solvable but not in radicals
But some are, and this one is.
Me, being an engineer:
Ok... google, what was the formula for Newton's method again?
...calculator goes brrr...
Hmmm, something around -1.325. Lets put -1.5, it will do its job.
Hahaha i did the exact same thing took a couple seconds.
Ahh it's around -1 , close enough. If we'd worked together we would have averaged it and nailed it.
Although there is not an analytical expression for the solutions of quintic equations, there are conditions on the existence of closed expressions for certain coefficients. Those I think were developed by Abel, but I'm not quite sure.
He is a Smart guy ,j would like to have brain on this level
- 1,324718
My procédure was different, based on complex numbers.
I differentiated once, twice, etc. and observed there was only one real root. So there are complex roots.
Then I tried the assumption that two of the complex ones were on the unit circle, i.e. x=exp(+/- i thêta).
In order for exp(i 5theta) +exp(i 4theta) to equ)al -1, aka exp(i π), we need
sin(4theta) = - sin(5theta) and cos(4theta) + cos(5theta) = -1
Implying cos(4theta) = cos(5theta) = - 1/2
Which (a graphic is useful here) gives
theta = +/- (2/3)π = +/- 120°
or
x = (1/2) +/- i (V3/2)
or, combined,
x²+x+1=0.
The assumption proved correct.
Dividing the original equiation by this factor leads to the cubic solving following your demonstration.
Calling the two cube roots a and b, we have
x0 = a+b
and
x1,x2 = (a+b)/2 +/- i |a-b|/2
Note that x^3=1 has three solutions: 1, w, w^2, we have w^5+w^4+1=w^2+w+1=(w^3-1)/(w-1)=0 and w^10+w^8+1=w+w^2+1=(w^3-1)/(w-1)=0. Therefore w and w^2 are the solutions of the function x^5+x^4+1=0. Thus the polynomial x^5+x^4+1 has factor x^2+x+1. One could simply conclude that x^5+x^4+1=(x^2+x+1)(x^3-x+1)
who else thought it was -1 and then found out that there is still one remaining?
I have a simpler way, you can derive the equation and find the extrim point to be x=0,x=-0.8 , by ploting it you see that at x=-0.8 you get y=0 ( a solution) at x=0 you get y=1, it is also noticible that the derivative of the function is 0 at this points and positive elsewere, that meens that this is a monotoic function and the only solution is at x=-0.8
but the answer is -1.3247
I would just use the Newton-Raphson method, lol.
Engineer?
I would have cranked the equation into an online polynomial solver, like Wolfram, it would give in two seconds the five solutions as stars in a nice complex plane. By the way, do you know that the family of polynomials equations: x-1=0, x^2+x-1=0, x^3+x^2+x-1=0, .... are related with the Fibonacci numbers, and of course with the Golden Ratio??
Newton's method should converge on a solution in about 5 iterations. I haven't tried it though.
@@jimgraham6722 :: The number of iterations for convergence for Newton Rapson depends on the seed value you give in as Xo, and the precision you need. And also not to fall in a zone where the function has a valley, where the iterations will go up and down, oscillating.
@@powerdriller4124 Yes indeed, NM has some limitations, but to me it is fun to watch it resolve some otherwise difficult problems using a cheap programmable calculator.
That was a brilliant solution, awesome videos!!!
Never clicked so fast!
Me too
Same xD
#nerds 🧮
Wow what a magnificent question. I am a teacher from the interior of Brazil and I am surprised by this question
this vedio so good to help crack my exam next wek its iit jee advanced sir can you pls help thank you sir
This guy is a genius I swear.
Good to see that you decided to grow your beard😊😊😊
Substituting x with 1 0 -1 -2
the value lies between -1 and -2
me: searching how to solve quadratic equations...
he: lets solve a 5th equation!
me: ... *head bangs against the table" ...
he also showed it for cubic but not the quadratic one. :D
One solution is cube root of unity that is -1+√3i/2,-1-√3i/2 try putting them by x^3=-1
According to complex comcepts w,w^2 are roots of unity and w^3=1 and 1+w+w^2=0
Therefore the equation x^5+x^4+1=0 becomes x^2.x°3+x.x^3+1=0 now x^3=1 for x=w therefore equation becomes 1+w+w^2 HP
How were people here before it came out
I put my unlisted videos in my playlists
blackpenredpen ohhhhh, thanks
@@blackpenredpen you could just graph ut aswell
Theta function can also be used to solve quantic and sixtic polynomials, however, for septic polynomials the Segal function is required.
As soon as i saw the equation i instantly calculated the solution -1.3247 in my head no problem.
I am so clever that I instantly calculated the answer in your head and made you think it was all your idea 😂🤣😂🤣.
You're either lying or exagerating
Purely amazing!
That's nothing. I calculated the answer in BPRB's head.
You mean you calc that with brain calculator?
Equation with only 1, -1, and 0 : try complex roots of 1, say a , and 1/a must be solution so we get a new polynomial equation: 1+a+a^5=0. the difference yield the equation : a^4-a=0,
a(a-1)(a^2+a+1)=0 : and try factoring the original equation by a^2+a+1.
1 real root
4 imaginary root
By d carties law of sighn
This is really genius. All is derived in a logical and understandable manner. I tried to figure out the polynomial multipöication but it got to complicated. Still I had the idea that maybe x^2 + 1 or x^2 + x + 1 could be the quadratic factor, since both of these end with 1 and have no real solution. But the grouping scheme was great and much easier!
Given that there is only one real non-repeated solution, this means it has to come from the cubic factor and not from the quadratic. This means that the cubic factor is a case where Cardano's formula works directly, because the cubic discriminant will be positive for the single non-repeated root. The quadratic factor could give us two real roots, or a real/repeated root, but not just one real/distinct root.
If there were only one real root and it were repeated, it could come from the quadratic factor. This isn't possible with an odd degree polynomial, since a repeated root means there's a turning point at the root. It has to turn back around and cross the x-axis again. So any odd-degree polynomial with a real repeated root, will have another distinct root as well.
Why can't B and E be like 4 and 0.25 for example? 2:08
We can also use the concept of primitive root of unity. Suppose that w^3 = 1; w is the primitive cube root of unity =/ 1. So w^5 + w^4 + 1 = w^3 + w^2 + 1 => (x^2 + x + 1) is a factor.
i just know that evryone thought "hm ok so, it's not zero" first thing first ahah
Also, it's not -1 either
That's what I thought
@@anshumanagrawal346 Is it greater or smaller than -1 !
whenever u have higher degree polynomials just check ones iota negative iota cube root of unity and square of cube root of unity and negatives of it.....
i just checked cube root of unity and square of it was the solution of this equation. So i just divided the polynomial by x^2 + x + 1 and it will be divisible because zeroes of this polynomial are cube root of unity and its squared and the actuall polynomial zeroes were also the same, so after dividing we just have a cubic equation to solve.......
8:55 uhhhh no dont do that
Here is another trick to find the factorization of the polynomial. Let a be a primitive third root of unity. Hence a^3=1 and a^2+a+1=0. Substituting x=a in the given equation we get a^5+a^4+1=a^2+a+1=0. Hence both the primitive third roots must be roots of the given polynomial also. Hence x^2+x+1 must divide the given polynomial. The other factor is obtained easily by long division.
meanwhile, physics student type the equation into mathematica
Lol
Similar factorable quintic polynomials with ±1 coefficients: x⁵-x⁴+x³+1, x⁵+x+1, x⁵-x⁴-1, x⁵-x²-x-1, x⁵+x²-x+1, x⁵+x⁴+1, x⁵+x-1, x⁵+x⁴+x³-1
I don't know but can anyone explain why there's a "±1" in the end of every existing equation out there 😂
Square root can be from either positive or negative number squared
Love it hat you've mentioned the Cardano formula at least for the reduced cubic👍
I saw one time ago that every polynomial in the form of x^3a + x^(3b+1) + x^(3c+2) is multiple of x²+x+1, so you can divide them for x²+x+1 and solve.
Thats so cool
How is the given quintic in this form? Are a,b and c positive integers? You did not state the "rule" clearly.
@@calvinjackson8110 I implicitely understand that yes, they are positive integers. Here, a=0 b=1 c=1 gives the polynomial in the video
Note that both terms inside the cube roots are negative so if you are trying to calculate this numerically it might be useful to express them as the negative of the cube root of their absolute value otherwise you might end up picking one of the complex cube roots _i.e._ -((1/2 - sqrt(23/108))^(1/3)) - (1/2 + sqrt(23/108))^(1/3) ~ 1.32472
I could see that "w and w^2" are the solutions of the original equation, coz w^2 + w + 1 = 0 and w^3 = 1. So I got x^2 + x + 1 as a factor directly. Works in this particular case only though 😅
Divide through by x^4, get the equation(1) x = -1 -1/x^4; Guess that the root must be negative, because x^4 is always positive. observe that x^5 - x^4 = -1. Guess x = -1.2; now iterate (1) starting with x= -1.2 and the answer will eventually converge to -1.324717957.....You can improve the rate of convergence by averaging the input and output numbers each time..
That solution, -1.3247, is the negative of the famous plastic ratio given by x^5 - x^4 - 1 = 0, the ratio constant of the Padovan sequence with the recurrence relation A(n) = A(n-2) + A(n-3). What famous ratio solves this quintic: x^5 - x^3 - x^2 - x^1 - x^1 - x^0 = 0? (Note the sequence of exponents).
if w is the complex cube root of unity ... it is obvious both w and w^2 are roots of x^5+x^4+1 ..so it is obvious that (x-w)(x-w^2)=x^2+x+1 is a factor.. A similar logic applies when you are asked to determine if weird numbers like 1280000401are prime or not. The trick is to see if the number can be written as a polynomial x^(2 mod 3) + x^( 1 mod 3) + 1 ...then by the above logic it has to be divisible by x^2+x+1 ... for example the number above is 20^7+20^2+1 .. hence divisible by 421=20^2+20+1.
WOW , it's awesome , Why didn't you complete the solution ? !!! Where is the 2nd part ???
Am I the only one who went straight to desmos? It's fascinating how you tackle it!
You make me love math. Thank you!
We solve as math teachers cubics as well
You might look at some examples
th-cam.com/video/ircZ_M1m3I8/w-d-xo.html
0:45 I don't have a formula for the general case, but at least I figured out that by proceeding in the same way as in the video, one can factor x^5 + ax^4 + a^5 into (x^2 + ax + a^2) (x^3 - a^2 x + a^3).
To solve this équation it possible to write two fonctions y = x^4 and Other fonction y = - x^5 - 1 and built their curves and the solution is the point sur of their intersection .
The 2nd degree eq solution is really interesting its - e^pi/3