I have another math question, and it's if this introduction to polynomials I wrote for myself is any good. The things in parentheses like (positive whole numbers) are for later if I come back to do a quick read-through given my less than stellar memory.
A polynomial (from the Greek word poly, meaning many, and the Latin nomial, meaning names or terms) is an algebraic expression which consists of variables (also referred to as indeterminates) and coefficients that only involves the operations of addition, subtraction, multiplication, and exponentiation to nonnegative integer powers with a finite amount of terms. In a way, you could say it is simply a thing that is composed of a sum of terms that are made up of coefficients multiplying variables raised to the power of positive whole numbers or zero (nonnegative integers). This can still be considered correct, because adding a negative number is equivalent to subtracting its positive counterpart. In fact, in the expression 5 - 2y, the coefficient of the second term is said to be negative 2 rather than 2.
Exponents and variables do not have to be explicitly shown in every polynomial, as in 5 + 4 - x, the terms 5 and 4 are constants (terms with no variables in this case), but they are each implied to be the coefficients of x⁰ and anything to the zeroth power is one, making them each multiply by one, therefore being 5 + 4. Lone variables are implied to have a power of 1, so the term x is actually x¹, meaning this expression is actually 5x⁰ + 4x⁰ - x¹, but we do not write out x⁰ or x¹ because it would be redundant.
Feb 03 · 3 months ago
11 Comments ↓
Way too complicated for an intro, the mere mention of exponentiation conjures horrors from the depths of math analysis, and to rub in it we specify that it's to non-negative integer powers. Even if you are aware of all the other kinds of exponentiation you really want to forget it all when working with polynomials, so don't even go there. It's just addition and multiplication, every super complicated expression with just those operations you can simplify to a sum of multiplications so that's it, that's a polynomial. The powers in this context are just a shorthand for multiplication x*x = x^2, there is nothing you can have there other than natural numbers, so don't even mention it, it's natural. You don't even need to define x^0 or explain why we don't write x^1 or claim an implied x^0 in every constants, which is very confusing, since polynomials can have any number of unknowns/variables. In some other context where you restrict yourself to a single variable, then try to employ some formulaic machinery to prove some result or describe some algorithm, you might need to define x^0=1 to make your formula simpler and not have to explicitly address that edge case, and thus imply an invisible x^0 on all constants, but in an intro that's entirely unnecessary. It's important to do these things when and if necessary so that the reader understands the purpose, and doesn't ask themselves philosophical questions like why multiplying something 0 times yields 1, it's that way so that that one formula works, and there is no better use for that expression, no deeper meaning there.
Same about specifying "finite amount of terms", all that does is make me think about infinite number of terms and seethe.
Your summary is largely correct, though I must admit it's more advanced than a first inroduction needs to be.
One note: the kinds of polynomials you'll see in introductory algebra are finite, but polynomials can have infinitely many terms. Polynomials with a finite number of terms can even be written to have infinitely many terms by multiplying all higher-degree terms by 0. For example, -1 + 3x - 3x^2 + x^3 = -1 + 3x - 3x^2 + x^3 + 0x^4 + 0x^5 + 0x^6 + ... . (That may seem pedantic, but it becomes important once you get to calculus.) You probably don't need to worry about that for now.
🗡️ The_Jackal [OP] · Feb 03 at 15:48:
@namark Well, I had written this as an intro for myself alone. I've been struggling with math my entire life so I wanted to make sure I could put in as much info as I could for myself (I am far, far behind on math. I never really made it to high school math) and still understand. I also wrote it that way because when looking around for exactly what polynomials were and in most of the definitions I found they would describe them like that, including the finite number of terms part. Would describing a polynomial as an algebraic expression that can't have any negative numbers, fractions, mixed numbers, or variables as exponents or division by a variable be a better explanation?
I guess a better question just would have been if I was correct, rather than if it would've been good for beginners.
🗡️ The_Jackal [OP] · Feb 03 at 15:56:
@jsreed5 Oh, thank you for clarifying. I made it this advanced because given my memory I wanted to put as much info as I could for myself that I could handle in that introduction and clear things up for myself I might wonder if I come back to it, like the definitions listing it as 'coeffecients multiplying variables exponentiated by nonnegative integers'. The implied x⁰ and x¹ made sense to me, so I included it for myself. I'm writing them like this because I should be half way through college, but I don't even have a GED yet (I never made it past 8th grade math and I'm wondering how far I even went into that), and I want to be an electrical engineer and maybe look into some chemistry. That's a LOT of catching up to do.
With that said, do you have any suggestions on how I could get what I wrote across but more concise?
👻 darkghost · Feb 03 at 16:12:
I have some chemistry in my background. You don't need very advanced math to do it for the most part. Algebra gets you far, statistics helps, and things like calculus can help with things like rates and integration for areas under curves, but once you get to calculus you're using a computer for everything anyways.
🗡️ The_Jackal [OP] · Feb 03 at 16:33:
@darkghost I had wanted to try getting some nuclear physics alongside the electrical engineering, but I doubt there's colleges or universities around me that would offer that, so I went back to the idea of chemistry. This isn't all just because I think I might get a high paying job, I got tired of not really knowing anything and wanted to 'pursue knowledge'. Specifically with a main focus in digital electronics like computers, all about the hardware, software, etc. and maybe something like chemistry or physics. Studying reality itself would be interesting to me. I'm also trying to build a foundation in math for all of this, and so far Khan Academy and random bits of searching on the topics are the best I have right now. I know some good textbooks would be better. I just hope this can prepare me for the GED, because math was always my weakest skill.
👻 darkghost · Feb 03 at 17:31:
Understanding how things work is the number one way to avoid being scammed. But it helps in everyday life as well. A study course focused on the GED will be most beneficial for passing the GED and the other skills will help you beyond it. After the GED, some leading universities have free courses online such as Stanford and Harvard with the option of paid certificates of completion. They're not degree programs but they'll help level up skills, maybe help you find a passion. My 2 cents as an internet stranger.
🗡️ The_Jackal [OP] · Feb 03 at 17:38:
@darkghost Thank you. I do have a book on the GED and found a website for getting ready for the GED, but I'm so far behind in math I really should catch up with something like Khan Academy and take notes like I'm doing beforehand. Speaking of calculus, it's like how I'd always use a calculator for something like 343 times 567. I'll use a calculator, but I still want to understand the concept behind it and probably be able to do it with enough time if I did it on paper/didn't have a calculator or program for it. I'd also like to try keeping a notebook for myself that goes as indepth as I can in a way I can understand for myself, and then probably use other notebooks to have notes that might simplify them out for me, but that's for much later. I only have the one book for now, but it's messy and will definitely need redoing in anotner book later. My main worries when I write a note like I did in this post is if I just got it blatantly incorrect or got a small detail wrong that could bite me in the ass later.
🗡️ The_Jackal [OP] · Feb 03 at 23:37:
@jsreed5 I wondered after this, if I were to correct the part about finite terms, would there be anything left that's incorrect? I went ahead and corrected the part about 'coeffecients of variables raised to powers' but left the section regarding x⁰ and x¹ in for myself if I come back confused after sometime and see that as a definition before looking at my notes, because when I first searched I would see definitions like that. So far if I were to summarize it, would 'A polynomial is an algebraic expression which is a sum of terms that only contain the operations of addition, subtraction, multiplication, or exponents. It cannot have negative numbers, variables, fractions, or mixed numbers as exponents nor include the radical sign (square root symbol) or division by a variable' be correct? Excluding a short introduction to degrees, at least. After doing some looking around, I found something about the 'power series' which I heard had things called polynomials but weren't quite polynomials like the ones you're introduced to in introductory algebra.
omg no, don't listen to the infinite guy, polynomials are finite, for the infinite stuff they have a separate name "power series", cause those abominations are so far removed from actual polynomials that it's not funny anymore.
Your stuff is totally correct, I just meant that it's too much information for an intro, like if you were writing a book or something, but if it's notes for an exams it all makes sense. Exams test breadth instead of depth, otherwise most people would fail them and the all them institutions would make no money.
Polynomials are extremely basic, and you can define them with just addition and multiplication and that's all you need to understand them. If it were me, I might've even defined multiplication to really drive the point home how basic we are talking. Now actually putting that on paper in a way you could fight an examiner over its correctness, that's some hard work (and you would need to fight, cause you would get bashed just for going out of line), so yeah, better stick with what you've got there.
👻 darkghost · Feb 04 at 10:12:
Agreed polynomials are pretty simple. They're purposely scope limited to be the format of (number)(variable)^(integer exponent) + (number)(variable)^(integer exponent) + (repeat this pattern a finite number of times)
2x + 4x² + 5x³ + 2x⁴ + 9x⁵ + 5x⁶ - 3x⁷ - x⁸ is an example.