A Nice Problem from Chris Juravich | Algebraic Expressions

🤩 Hello everyone, I'm very excited to bring you a new channel (aplusbi)
Enjoy...and thank you for your support!!! 🧡🥰🎉🥳🧡
/ @sybermathshorts
/ @aplusbi
Original post: / 1707868884903072077
⭐ Join this channel to get access to perks:→ bit.ly/3cBgfR1
My merch → teespring.com/stores/sybermat...
Follow me → / sybermath
Subscribe → ua-cam.com/users/SyberMath?sub...
⭐ Suggest → forms.gle/A5bGhTyZqYw937W58
If you need to post a picture of your solution or idea:
intent/tweet?text...
#algebra #algebraicexpressions
via @UA-cam @Apple @Desmos @NotabilityApp @googledocs @canva
x^2+2x=1, x^5-1/x^5=?
An Exponential Equation | 2^{3^{5^x}} = 3^{2^{5^x}}: • An Exponential Equatio...
A Surprising Exponential Equation: • A Surprising Exponenti...
PLAYLISTS 🎵 :
Number Theory Problems: • Number Theory Problems
Challenging Math Problems: • Challenging Math Problems
Trigonometry Problems: • Trigonometry Problems
Diophantine Equations and Systems: • Diophantine Equations ...
Calculus: • Calculus

[@chrisjuravich3398]

Thank you for the mention, sir! Nice solution. ❤
My solution method takes advantage of a specific property of the Lucas sequence. I’ll post my solution here and over on X later today.
And, yes, you pronounced my name exactly right. 😊

[@SyberMath]

No problem! Thank you for the ideas 🤩
Looking forward to your post!

[@iMíccoli]

Nice I got the right answer. So I used the second method but I divided both sides by x/x² to get x-1/x=-2 but it's pretty much the same thing.

[@demenion3521]

My first instinct with this type of problem is Vieta's formula. Note that from the quadratic we get two solutions, let's call them a and b, such that ab=-1 and a+b=-2. Now that means that 1/a^5=-b^5. Then we can solve the problem using the sum of 5th powers formula.
a^5+b^5 = (a+b) (a^4+b^4-ab(a²+b²)+(ab)²)
= -2((a²)²+(b²)²+(a²+b²)+1)
= -2((1-2a)²+(1-2b)²+(a²+b²)+1)
= -2(3-4(a+b)+5(a²+b²))
= -2(3+8+5((a+b)²-2ab))
= -2(11+5(4+2))
= -2(41)
= -82
From the second to the third line I used the original equation to get x⁴ as in the third method in the video. From the 4th to the 5th line I completed the square, but using the equation again would've done the same.

[@demenion3521]

If you don't want to use the original equation at all except for the sum and product of the roots, you can also simplify the sum of 4th powers by completing the square as
a⁴+b⁴=(a²+b²)²-2(ab)²
=((a+b)²-2ab)²-2
=(4+2)²-2
=34

[@mcwulf25]

I expanded (x+ 1/x)^5.
We only need the odd powers. So I also expanded (x+1/x)^3 and this gave me the answer in terms of (x+1/x).

[@trumpetbob15]

I attempted the third method but gave up on the long division step; however, after checking my work with a calculator, I would have gotten the correct answer so I'll take that as a win.

[@artfifi5424]

thqnk you it is so funny I like it ❤

[@robertingliskennedy]

👌

[@scottleung9587]

I used the third method.

[@rakenzarnsworld2]

Answer: -9

[@alextang4688]

I use 3rd method. 😋😋😋😋😋😋

[@SyberMath]

Great 👍

[@goldfing5898]

So what is your 3rd method?

[@SyberMath]

express all powers in terms of x linearly and sub

[@goldfing5898]

@@SyberMath This is way to vague for me, but I currently have no time to calculate it myself.

[@trojanleo123]

Does your colleague have a UA-cam channel? Sorry I don't use Twitter. I find it too toxic and I'm also not a Elon Musk fan.

[@chrisjuravich3398]

I don’t have a UA-cam channel. I use Twitter for math and baseball related topics, mostly math. It’s a great place to interact with folks with a common interest, if you can ignore the noise. Like UA-cam, you can pick and choose what you consume.

[@neuralwarp]

You have x⁵ + ... - x(‐⁵) so isn't x-1 a factor?

[@goldfing5898]

My method:
x^2 + 2x = 1
Divide by x:
x + 2 = 1/x
So
x^5 - 1/x^5
= x^5 - (1/x)^5
= x^5 - (x + 2)^5
Substitute t = x + 1, so x = t - 1 and x + 2 = t + 1:
...
= (t - 1)^5 - (t + 1)^5
= (t^5 - 5t^4 + 10t^3 - 10t^2 + 5t - 1)
- (t^5 + 5t^4 + 10t^3 + 10t^2 + 5t + 1)
= - 10t^4 - 20t^2 - 2
= -2 * (5t^4 + 10t^2 + 1)
Since x^2 + 2x = 1
x^2 + 2x + 1 = 2
(x + 1)^2 = 2
With t = x + 1, we get
t^2 = 2
t^4 = 4
and finally
...
= -2 * (5*4 + 10*2 + 1)
= -2 * (20 + 20 + 1)
= -2 * 41
= -82
And this is the final result.

[@SyberMath]

you can edit

[@goldfing5898]

@@SyberMath I was (as so often) out of home when typing, and for some obscure reason, the Edit and Delete function does never work on my smartphone when doing youtube.