#polynomials — Public Fediverse posts

Alright, future engineers!
**Factoring:** Breaking a polynomial into simpler expressions (factors) that multiply back to the original.
Ex: `x^2 - 4 = (x-2)(x+2)`.
Pro-Tip: Always look for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Breaking a polynomial into simpler expressions (factors) that multiply back to the original.
Ex: `x^2 - 4 = (x-2)(x+2)`.
Pro-Tip: Always look for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Breaking down an expression into a product of simpler ones (its factors).
Ex: `x^2 - 4 = (x-2)(x+2)`.
Pro-Tip: Always look for a GCF first! It simplifies everything. Makes solving equations much easier.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Breaking down an expression into a product of simpler ones (its factors).
Ex: `x^2 - 4 = (x-2)(x+2)`.
Pro-Tip: Always look for a GCF first! It simplifies everything. Makes solving equations much easier.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Remainder Theorem:** When you divide a polynomial P(x) by (x-c), the remainder is P(c).
Ex: P(x)=x^2+1. Divide by (x-2), Remainder P(2)=5.
Pro-Tip: If P(c)=0, (x-c) is a factor! Quick way to test roots.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Remainder Theorem:** When you divide a polynomial P(x) by (x-c), the remainder is P(c).
Ex: P(x)=x^2+1. Divide by (x-2), Remainder P(2)=5.
Pro-Tip: If P(c)=0, (x-c) is a factor! Quick way to test roots.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Breaking down an algebraic expression into a product of simpler ones.
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always look for a Greatest Common Factor (GCF) first, it simplifies everything!
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Breaking down an algebraic expression into a product of simpler ones.
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always look for a Greatest Common Factor (GCF) first, it simplifies everything!
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Breaking a polynomial into simpler expressions (factors) that multiply to the original.
Ex: `x^2 - 4 = (x-2)(x+2)`
Pro-Tip: Always check for a GCF first! It simplifies everything.
#AlgebraSkills #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Rewriting a polynomial as a product of simpler expressions (factors).
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always look for a GCF first! It simplifies everything, making further factoring easier.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring:** Rewriting a polynomial as a product of simpler expressions (factors).
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always look for a GCF first! It simplifies everything, making further factoring easier.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!

**Factoring:** Breaking a polynomial into simpler expressions that multiply to the original.
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always pull out the GCF (Greatest Common Factor) first! It makes subsequent factoring easier.

#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!

**Factoring:** Breaking a polynomial into simpler expressions that multiply to the original.
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always pull out the GCF (Greatest Common Factor) first! It makes subsequent factoring easier.

#Algebra #Polynomials #STEM #StudyNotes

Factoring "short-sleeve" RSA keys with polynomials

https://blog.trailofbits.com/2026/06/12/factoring-short-sleeve-rsa-keys-with-polynomials/

#HackerNews #factoring #RSA #cryptography #polynomials #cybersecurity #hacking

Factoring "short-sleeve" RSA keys with polynomials

https://blog.trailofbits.com/2026/06/12/factoring-short-sleeve-rsa-keys-with-polynomials/

#HackerNews #factoring #RSA #cryptography #polynomials #cybersecurity #hacking

Alright, future engineers!
**Factoring** breaks down a polynomial into simpler multiplied terms.
Ex: `x^2+5x+6 = (x+2)(x+3)`
Pro-Tip: Always check for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring** breaks down a polynomial into simpler multiplied terms.
Ex: `x^2+5x+6 = (x+2)(x+3)`
Pro-Tip: Always check for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Synthetic Division** is a shortcut to divide a polynomial by a linear factor `(x-k)`.
Ex: `(x^3 - x^2 + x - 1) / (x-1)` use `k=1`.
Pro-Tip: ONLY works if the divisor is `(x-k)`! Not for `x^2+1` or higher powers.
#Polynomials #Algebra #STEM #StudyNotes

Alright, future engineers!
**Synthetic Division** is a shortcut to divide a polynomial by a linear factor `(x-k)`.
Ex: `(x^3 - x^2 + x - 1) / (x-1)` use `k=1`.
Pro-Tip: ONLY works if the divisor is `(x-k)`! Not for `x^2+1` or higher powers.
#Polynomials #Algebra #STEM #StudyNotes

Alright, future engineers!
**Factoring** breaks down a polynomial into a product of simpler expressions.
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always look for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!
**Factoring** breaks down a polynomial into a product of simpler expressions.
Ex: `x^2 + 5x + 6 = (x+2)(x+3)`
Pro-Tip: Always look for a Greatest Common Factor (GCF) first! It simplifies everything.
#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!

**Factoring** breaks a polynomial into simpler expressions (factors) that multiply to it. Ex: `x^2+5x+6 = (x+2)(x+3)`. Pro-Tip: Always look for a Greatest Common Factor (GCF) first!

#Algebra #Polynomials #STEM #StudyNotes

Alright, future engineers!

**Factoring** breaks a polynomial into simpler expressions (factors) that multiply to it. Ex: `x^2+5x+6 = (x+2)(x+3)`. Pro-Tip: Always look for a Greatest Common Factor (GCF) first!

#Algebra #Polynomials #STEM #StudyNotes

I've been making plots of the sets of roots of some polynomials. They look more interesting than I'd expected! And I've been making sounds from them, too. I have a bunch on this page of my website (I'll probably be adding more). Click on the images to embiggen them, and if you're in a hurry, the last sound on the page is the most catchy. https://www.madandmoonly.com/doctormatt/sound/littlewoodPolynomials/

#mathematics #math #maths #polynomials #sonification #illustration #sound

I've been making plots of the sets of roots of some polynomials. They look more interesting than I'd expected! And I've been making sounds from them, too. I have a bunch on this page of my website (I'll probably be adding more). Click on the images to embiggen them, and if you're in a hurry, the last sound on the page is the most catchy. https://www.madandmoonly.com/doctormatt/sound/littlewoodPolynomials/

#mathematics #math #maths #polynomials #sonification #illustration #sound

A now a message from the Society for the Protection of Polynomials.

Hi! We at the Society for the Protection of Polynomials remind you that factoring a polynomial is not a harmless operation. Whenever you factor a polynomial, you are causing untold damage to it.

Don't factor polynomials.

Keep them whole.

This was a message from the Society for the Protection of Polynomials.

#polynomials #SocietyForTheProtectionOfPolynomials

🤓 Ah yes, the riveting tale of #Lagrange Interpolating Polynomials—a math nerd’s wet dream! 📈 Just what the internet needs: *another* 5000-word novel on fitting curves to dots! 🤯 Because who doesn’t love a good bedtime story about polynomial coefficients? 🙄
https://eli.thegreenplace.net/2026/notes-on-lagrange-interpolating-polynomials/ #mathnerd #mathstories #polynomials #datafitting #curvefitting #HackerNews #ngated

🤓 Ah yes, the riveting tale of #Lagrange Interpolating Polynomials—a math nerd’s wet dream! 📈 Just what the internet needs: *another* 5000-word novel on fitting curves to dots! 🤯 Because who doesn’t love a good bedtime story about polynomial coefficients? 🙄
https://eli.thegreenplace.net/2026/notes-on-lagrange-interpolating-polynomials/ #mathnerd #mathstories #polynomials #datafitting #curvefitting #HackerNews #ngated

Notes on Lagrange Interpolating Polynomials

https://eli.thegreenplace.net/2026/notes-on-lagrange-interpolating-polynomials/

#HackerNews #LagrangeInterpolate #Polynomials #Math #Education #DataScience #Algorithms

Notes on Lagrange Interpolating Polynomials

https://eli.thegreenplace.net/2026/notes-on-lagrange-interpolating-polynomials/

#HackerNews #LagrangeInterpolate #Polynomials #Math #Education #DataScience #Algorithms

#Noisevember! I revisited the Littlewood polynomial sound from day 2 of Noisevember. I thought to investigate a different sort of polynomial. Here, instead of polynomials with coefficients all ±1, the polynomials have coefficient ±1/(n+1) on the x^n term. As before, all roots of all such 15th degree polynomials are considered. (I really should create a gallery of these root plots so we can easily compare them.) Along the way, I realized I was making an error with the way I created "random" stereo that introduced a bunch of unneeded noise. So that's something! I'll have to go back and replace the Littlewood polynomial sound. https://soundcloud.com/matthew-m-conroy/out-keep1

Here's a plot of the roots (essentially the spectrogram of the sound).

#noise #sound #audio #math #maths #mathematics #polynomials #roots

#Noisevember! I revisited the Littlewood polynomial sound from day 2 of Noisevember. I thought to investigate a different sort of polynomial. Here, instead of polynomials with coefficients all ±1, the polynomials have coefficient ±1/(n+1) on the x^n term. As before, all roots of all such 15th degree polynomials are considered. (I really should create a gallery of these root plots so we can easily compare them.) Along the way, I realized I was making an error with the way I created "random" stereo that introduced a bunch of unneeded noise. So that's something! I'll have to go back and replace the Littlewood polynomial sound. https://soundcloud.com/matthew-m-conroy/out-keep1

Here's a plot of the roots (essentially the spectrogram of the sound).

#noise #sound #audio #math #maths #mathematics #polynomials #roots

Hi fam, have a fulfilling weekend! Some joker marked my account, e.g. this post about #polynomials & #calculus, which is a textbook example of #inclusion, as spam. This is incredibly unfair; So I'm asking you to share this: bsky.app/profile/paul... (w #ALText) as a protest. #education #mathematics

RE: https://bsky.app/profile/did:plc:omyr27fmzj3phbagch4sqyub/post/3lbwoytxzo22d

Convolutions, Polynomials and Flipped Kernels

https://eli.thegreenplace.net/2025/convolutions-polynomials-and-flipped-kernels/

#HackerNews #Convolutions #Polynomials #FlippedKernels #MachineLearning #DataScience

Convolutions, Polynomials and Flipped Kernels

https://eli.thegreenplace.net/2025/convolutions-polynomials-and-flipped-kernels/

#HackerNews #Convolutions #Polynomials #FlippedKernels #MachineLearning #DataScience

@catselbow
Astonishing! I hadn't come across this before - absolutely fascinating!
#maths #mathematics #polynomial #polynomials #CatalanNumbers #combinatorics #Galois #GaloisTheory
https://www.tandfonline.com/doi/epdf/10.1080/00029890.2025.2460966?needAccess=true

**A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode**

“_....the hyper-Catalan numbers 𝐶𝐦 count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type 𝐦), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation._”

Wildberger, N. J. and Rubine, D. (2025) ‘A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode’, The American Mathematical Monthly, pp. 1–20. doi: https://doi.org/10.1080/00029890.2025.2460966.

#OpenAccess #OA #Article #DOI #Maths #Mathematics #Math #Algebra #Polynomials #Academia #Academics

**A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode**

“_....the hyper-Catalan numbers 𝐶𝐦 count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type 𝐦), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation._”

Wildberger, N. J. and Rubine, D. (2025) ‘A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode’, The American Mathematical Monthly, pp. 1–20. doi: https://doi.org/10.1080/00029890.2025.2460966.

#OpenAccess #OA #Article #DOI #Maths #Mathematics #Math #Algebra #Polynomials #Academia #Academics

One day, one decomposition
A235034: Numbers whose prime divisors, when multiplied together without carry-bits (as encodings of GF(2)[X]-polynomials, with A048720), produce the original number; numbers for which A234741(n) = n

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A235034.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A235034.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #prime #divisors #polynomials #graph #threejs #webGL

One day, one decomposition
A235034: Numbers whose prime divisors, when multiplied together without carry-bits (as encodings of GF(2)[X]-polynomials, with A048720), produce the original number; numbers for which A234741(n) = n

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A235034.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A235034.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #prime #divisors #polynomials #graph #threejs #webGL

One day, one decomposition
A235033: Numbers which are factored to a different set of primes in Z as to the irreducible polynomials in GF(2)[X]

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A235033.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A235033.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #PrimeNumbers #irreducible #polynomials #graph #threejs #webGL

One day, one decomposition
A235033: Numbers which are factored to a different set of primes in Z as to the irreducible polynomials in GF(2)[X]

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A235033.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A235033.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #PrimeNumbers #irreducible #polynomials #graph #threejs #webGL

Math That Matters: The Case for Probability over Polynomials — https://anandsanwal.me/math-eduction-more-probability-statistics-less-calculus/
#HackerNews #MathThatMatters #Probability #Polynomials #Education #Statistics #Calculus

I ran across the Wikipedia article on Littlewood polynomials. It has a plot of all the roots of the degree 15 polynomials, that looks very nice. I thought I would create an animation showing the roots for degree 1, degree 2, etc. I also thought maybe I'd add a plot of the roots for something with degree higher than 15. Here is the degree 16 plot (this is reduced to 25% of the original image). It took 2 hours in Sage on my laptop, so I might try 17, even 18 - who knows? I have the thought that I ought to be able to reduce the precision, and this ought to speed things up a lot (since for plotting much lower precision than the default is needed). I don't particularly like blue, though: I'll have to try other colors.
https://en.wikipedia.org/wiki/Littlewood_polynomial

#mathematics #littlewood #littlewoodPolynomials #polynomials #plotting #sagemath #graphics #illustration

I ran across the Wikipedia article on Littlewood polynomials. It has a plot of all the roots of the degree 15 polynomials, that looks very nice. I thought I would create an animation showing the roots for degree 1, degree 2, etc. I also thought maybe I'd add a plot of the roots for something with degree higher than 15. Here is the degree 16 plot (this is reduced to 25% of the original image). It took 2 hours in Sage on my laptop, so I might try 17, even 18 - who knows? I have the thought that I ought to be able to reduce the precision, and this ought to speed things up a lot (since for plotting much lower precision than the default is needed). I don't particularly like blue, though: I'll have to try other colors.
https://en.wikipedia.org/wiki/Littlewood_polynomial

#mathematics #littlewood #littlewoodPolynomials #polynomials #plotting #sagemath #graphics #illustration

#Maths #Mathematics Truncated tetrahedron. An Archimedean solid, this polyhedron is mapped from the continuous polynomial ((x+y)/(z+1))^60+((y+z)/(x+1))^60+((z+x)/(y+1))^60 +(.6x+.6y+.6z)^60+(.6x+.6y-.6z)^60+(.6x-.6y+.6z)^60+(-.6x+.6y+.6z)^60-1=0 and comprises four regular hexagon and four equilateral triangle faces, twelve vertices and eighteen edges. #Polynomials

#Maths #Mathematics Truncated tetrahedron. An Archimedean solid, this polyhedron is mapped from the continuous polynomial ((x+y)/(z+1))^60+((y+z)/(x+1))^60+((z+x)/(y+1))^60 +(.6x+.6y+.6z)^60+(.6x+.6y-.6z)^60+(.6x-.6y+.6z)^60+(-.6x+.6y+.6z)^60-1=0 and comprises four regular hexagon and four equilateral triangle faces, twelve vertices and eighteen edges. #Polynomials

`We show that the Weierstrass method, like the well known #Newton method, is not generally convergent: there are open sets of #polynomials p of every degree d≥3 such that the dynamics of the Weierstrass method applied to p exhibits attracting periodic orbits.`

https://arxiv.org/abs/2004.04777

#rootFinding #optimizationTheory #optimization #computation #computing

@rwxrwxrwx
I wrote this function #'LAMBDAISE that turns a cl-buchberger:polynomial into an unevaluated lambda form at run time. I feel like this is going to have a more elegant expression, but I figure if
the lambdaiseing is happening offline it's okay. What do you think? What do other #CommonLisp #lisp users think? #polynomials will use for synth later

#100daystooffload on codes for turning symbolic polynomials into lambda forms
https://gopher.tildeverse.org/tilde.club/0/~screwtape/synthember-100days-tooffload/008.lisp.txt

@82mhz @thankfulmachine @AlgoCompSynth

So....#Lagrange #polynomials. They are pretty dang clever and pretty dang amazing. You can get an analytic (if that's the word I want) polynomial fit! That you can take a continuous derivative!

For arbitrary data, I can see why it might not work.

But for a physical object that actually is moving according to a 2nd-order kinematics (piecewise--really 3rd-order overall) but all you have data for the the first two orders, it might be a good way to recover the higher order(s).

#math #numerical

So....#Lagrange #polynomials. They are pretty dang clever and pretty dang amazing. You can get an analytic (if that's the word I want) polynomial fit! That you can take a continuous derivative!

For arbitrary data, I can see why it might not work.

But for a physical object that actually is moving according to a 2nd-order kinematics (piecewise--really 3rd-order overall) but all you have data for the the first two orders, it might be a good way to recover the higher order(s).

#math #numerical

Since I was teaching orthogonal #polynomials (OPs), I asked my class whether they’d read Great Expectations by #Dickens, which you may remember had APs :)