How to depress a cubic
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- Опубліковано 12 чер 2024
- The cubic formula is rarely used and rarely talked about. This video explains how to depress a cubic polynomial into a form that works with the cubic formula.
Watch Cube-root of Unity here:
• Cube Root of Unity
It is possible to generalize this method for quartic
(Euler did it)
To solve depressed cubic you can substitute
x = a+b
but to solve depressed quartic substitution would be
x = a+b+c
Let see what we will get after this substitution
x^4+px^2+qx+r
x = a+b+c
x^2 = a^2+b^2+c^2+2(ab+ac+bc)
x^2 - (a^2+b^2+c^2) = 2(ab+ac+bc)
(x^2 - (a^2+b^2+c^2))^2 = 4((a^2b^2+2a^2bc+a^2c^2)+2(ab+ac)bc + b^2c^2)
x^4 - 2(a^2+b^2+c^2)x^2+(a^2+b^2+c^2)^2 = 4(a^2b^2 + a^2c^2 + b^2c^2 + 2(a^2bc + ab^2c + abc^2))
x^4 - 2(a^2+b^2+c^2)x^2+(a^2+b^2+c^2)^2 = 4(a^2b^2 + a^2c^2 + b^2c^2 + 2abc(a+b+c))
x^4 - 2(a^2+b^2+c^2)x^2+(a^2+b^2+c^2)^2 = 4(a^2b^2 + a^2c^2 + b^2c^2) + 8abc(a+b+c)
x^4 - 2(a^2+b^2+c^2)x^2+(a^2+b^2+c^2)^2 = 4(a^2b^2 + a^2c^2 + b^2c^2) + 8abcx
x^4 - 2(a^2+b^2+c^2)x^2 - 8abcx + (a^2+b^2+c^2)^2 - 4(a^2b^2 + a^2c^2 + b^2c^2) = 0
So after comparing coefficients we have following system of equations
-2(a^2+b^2+c^2) = p
-8abc = q
(a^2+b^2+c^2)^2 - 4(a^2b^2 + a^2c^2 + b^2c^2) = r
This system of equations can be easily transformed into
Vieta formulas for cubic with the roots a^2 , b^2 , c^2
Although such generalization is possible i prefer to solve quartic by factoring into two quadratics
For depressed quartic undetermined coefficients
(x^2 - ax + b)(x^2 + ax + c) = x^4 + px^2 + qx + r
works nicely but when given quartic is not already depressed i prefer to use differece of squares first
@@holyshit922 sounds easyer
Why is it sad?
It fell into depression after the x² term left 😔
Next video on making the cubic cheerful please!!
How to depress a cubic? just show it a reality of life
Your claim at 17:20 that the other two complex conjugate roots of your depressed cubic
t³ + ²⁄₃t + ²⁰⁄₂₇ = 0
are obtained by multiplying the real root t = −²⁄₃ by ω = −¹⁄₂ + i·¹⁄₂√3 and by ω² = −¹⁄₂ − i·¹⁄₂√3 is incorrect, that is _not_ how it works. See my detailed comment on your previous video about solving depressed cubic equations.
In fact, what you should do to obtain the other two roots is multiply one of the two cube roots by ω and the other by ω² _in either order._
So, since we have
³√(−10 + 6√3) = −1 + √3
³√(−10 − 6√3) = −1 − √3
the roots of your cubic in t are
t₁ = ¹⁄₃((−1 + √3) + (−1 − √3)) = ¹⁄₃(−2) = −²⁄₃
t₂ = ¹⁄₃((−¹⁄₂ + i·¹⁄₂√3)(−1 + √3) + (−¹⁄₂ − i·¹⁄₂√3)(−1 − √3)) = ¹⁄₃(1 + 3i) = ¹⁄₃ + i
t₃ = ¹⁄₃((−¹⁄₂ − i·¹⁄₂√3)(−1 + √3) + (−¹⁄₂ + i·¹⁄₂√3)(−1 − √3)) = ¹⁄₃(1 − 3i) = ¹⁄₃ − i
I recently solved very interesting integral and want you to try it out
Integral of sqrt(x^2+a)
"a" is a constant
Isnt it done by integration by parts?
I think it will give a hyperbolic inverse, am I right? 🤔
There is Tschirnhaus transformation which allows to get rid of terms
x^{n-1} , x^{n-2},x^{n-3} from polynomial
hello first title sound intersting but maby the cubic needs mental help? xD
0:47 but if i can depress a cubic it has to be posible to get back the original form so in terms of the last video wouldn´t that be easyer ?
1:12 what is not posible?? how ever.
2:22 i think if you use the normal ruls of equantions it is unimportent wich number you choose it will work anyway!
3:19 ok i understand how it works but not whatfor you need it!
3:58 that way of using t i have seen it somewere befor and why you choose t and not z or r or what ever?
7:10 ok yes thats quite simple at this point
8:34 i think i schould learn the solution for (a+b)³
9:53 thanks for simplify it for me!
10:35 wouldn´t it be easyer to write (t/3) ?
11:28 i think i would had simplifyed the sum in the brackets before instead of writing it out but allright!
11:51 ok yes now it´s fine!
13:51 yes i remeber that! but anyway less you if you wan´t to use that
15:55 ok much fun to calculate that with out an calculator xD
16:34 oh ok now it looks posible
17:34 allright
17:41 an could you please explain me why do you add quotes at the end?
17:42 and please also explain me the quote it self!
yours sincerly
K.Furry
what if i want to make it happy
Can this method be still used if all the roots are real?
For example I have a cubic equation x³-10x²+31x-30=0
Then t³-7/3t-20/27=0 after this I am having trouble for computing t. Can anyone help me out?
Just divide original by x-5, no depresing
I prefer x=a+b , I find it easier to memorize.
(a+b)^3+(a+b)^2+(a+b)+1= a^3+a^2 * (3b+1)+a*(3b^2+2b+1)+(b3+b2+b+1)
We need the coefficient of a^2 to be zero, which means (3b+1)=0
b=-1/3
(a + b)^3 + (a + b)^2 + a + b + 1 = a^3 + (2 a)/3 + 20/27
why are the views so down nowadays? you are so underrated! hope everything gets better
Actually
Woah I never noticed the view counter ever, the videos he makes are SOO SOO SOO good regardless
Thats what UA-cam algorithm is sp unpredictable. There are still some math youtubers deserving more views including PN!
@@OrilliansGood to see you here
@@iqtrainer Do you know me from somewhere?
multiplying (x-1) on both sides taking
x ≠0
⇒ x= e^(2πi/4)
x= e^(πi/2)
i=1,2,3
x=i,-1,-i
Great video brother but I want to point out that you can complete a cube it’s analogous to completing the square. This was done over 500 years ago. Scipione Del Ferro and others have done it.
Yes. Read my comment on Prime Newton's video about solving the general depressed cubic equation x³ + px + q = 0.
Can you give proof of formula for root of cubical equation?
person named "Acubic":
Question: he said to multiply by the cube root of unity to get from -1 to the other two answers, what does he mean?
Don't bother, what Prime Newtons claimed is wrong. See my main comment on this video for an explanation how to do this correctly.
@@NadiehFan thank you kind viewer.
Still I would like to see the whole process, anв where those formulas for 2 other roots emerge (I've seen at least 2 variantsб maybe more). And, perhaps I missed smth, but i didn't get how we see how many roots there are. Also I'm wondering how Cardano (or Tartaglia) found all the three solutions without using complex numbers.
How to use cube root of unity to find all solutions ?
ua-cam.com/video/lHe6iieqzBw/v-deo.html
Suppose omega is principal cube root of unity
to get solutions you multiply one cubic radical by omega and the other one by omega^2
If you look at system which you get after solvig depressed cubic
p = 3ab , -q = a^3 - b^3
it would be clear
What about casus irreducibilis in my opinion it is worth considering this case
because after using de Moivre theorem you will get trigonometric solution from complex cubic radicals
Another UA-camr shows a cubic formula derivation of one complex conjugate from a pair root found in ax^3 + bx^2 + cx + d = 0 whether or not it was a real root from the more than one real roots possibility. Because a cubic polynomial is at least from a real root (x + r) multiplied to a quadratic polynomial, the cubic root formula is related to if the resulting quadratic polynomial had real two roots or two imaginary numbers. The more versatile cubic formula is that of ax^3 + bc^2 + cx + d root formula.
How in the world did you get minus 2
We have
³√(−10 + 6√3) = −1 + √3
³√(−10 − 6√3) = −1 − √3
so we get
³√(−10 + 6√3) + ³√(−10 − 6√3) = (−1 + √3) + (−1 − √3) = −2
if 6sqrt(3)-10 = (sqrt(a)-b)³ then you get 6sqrt(3) -10= (a+3b²)sqrt(a) - (3ab+b²).
You see immediately that a=3 (sqrt!) and after solving (a+3b²)=6 you get b=1.
So, (sqrt(a)-b)³ = (sqrt(3)-1)³ = 6sqrt(3) -10 and therfore sqrt(3)-1 = (6sqrt(3) -10)^(1/3)
The solution of (-6sqrt(3) -10) yields -sqrt(3)-1.
The sum (sqrt(3)-1) + (-sqrt(3)-1) = -2
I legitimately memorized both the cubic and quartic before but I think I memorized the quartic wrong bc the font of the thing I was looking at was too small
Misread as ‘how to make a child depressed’ and still clicked
This can be solved with factoring by grouping no need for depressing this cubic. The answers are i and minus i and -1.
I don't wanna cubic to be sad MONSTER😢
depress?? 😂
😂
I must be a cubic.
One way I never see is to set x = (t-b)/3a, which also eliminates the quadratic term but scales the cubic to only have integers as coefficients and a monic leading coefficient. The numbers are bigger but working with integers is much nicer than working with fractions.
You wrote that incorrectly. The denominator would be inside grouping symbols:
x = (t - b)/(3a).
Why are you making it sad? 😭😭
So cool sir, i like this content, so atractive.
Hum... Why ? 😭 I mean that's interesting but it's very complicated and there's an evident solution that you can find faster.
The Cardano's formula is really awful...
I commend you on not making a depressed cubic joke in this video. If humor is a low hanging fruit most people can't resist picking a fruit like that. I enjoyed the video, this was a new topic for me. Thank you!
Sir please make videos on how to get good marks in indian competitive exams like IOQM,RMO,INMO.
This video is .... depressing ;-)
First
Tell it its annoying and nobody likes it