I love math until it gets in to polynomials that add on top of each other, from slope intercept to the quadratic formula to Trig. It's usually in Trig that the "what's the point of it all" instinct kicks in, and it ends horribly. The basis is in polynomials and formulas using them, which starts in Algebra.
How does one get to enjoy polynomial equations? How does one feel good about doing them? I have no love for polynomials. No love at all.
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Anonymous2005-08-31 23:32
Newtonian (aka easy) physics involves a lot of polynomials. They're useful if you're considering how long it will take that bullet you just fired to hit the ground, for example.
More importantly, though, I think you're missing the point of mathematical education. Many direct aspects of your math education will never be used again, but that's not where the value lies (unless you end up in a career where you need it. Engineering, accounting, research of all kinds, etc).
The real value of math lies in your learning to approach a problem analytically and solve it. This is an *extremely* valuable skill, and surprisingly few people develop it. It allows you to understand how to approach real-life problems such as paying as little as legally possible in taxes, or figuring out what angle to cut the board on for your new deck.
I took 4 semesters of calculus in college. I've used the actual mathematical principles of integrals and differential equations only *very* rarely, but the techniques I learned for proving and experimentation have been useful time and again in my Software Engineering career.
Don't ever dismiss math as "useless". Remember, it's the *mode of thought* that you're learning, and that's a very difficult thing to learn when you get older.
Algebra is when math first starts becoming an abstract exercise. You're no longer dealing in trivial arithmetic, but rather with abstract concepts. Bend your brain around it, it's *way* worthwhile. Don't fall into the trap of "I'll never use this". Trig et al is some pretty basic, important stuff.
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Anonymous2005-09-01 9:26
In addition to what >>2 has already said, you have to realise that polynomial solving is a very basic skill in math. It's not only used in pure-algebra fields, but many times when you're doing any analytical math, you're likely to encounter polynomials and you simply need to be able to go through it in auto-pilot mode. Even if you're just doing analysis or number theory, you'll still occasionally encounter poloynomials in the course of a proof.
Bite the bullet and do lots of examples from bottom up is all I can offer. Eventually, you'll realise it's the same thing over and over again but with different symbols and maybe a few more steps.
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Anonymous2005-09-02 6:05
What are you talking about, polynomials are well fun!
If you're thinking of taking maths to a higher level, beware that there's something your teachers won't warn you of: it suddenly stops being about techniques for solving problems and becomes proofs. You have to learn to understand and remember other people's proofs, and to be able to prove things yourself. It's hard.
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Anonymous2005-09-02 13:20
I just need some way to visualize what a polynomial "means." I know that in some vague sense, it represents a line on a graph, but I don't see "how" just by looking at a polynomial. It's gibberish to me. I understand X = X coordinate, Y = Y coordinate, and somehow number next to a variable equals a slope.
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Anonymous2005-09-02 14:04
Get some graph drawing program to get a feeling what polynomal curves look like. It's all quite simple really.
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Anonymous2005-09-02 16:27
>>5
If you're talking about a complex polynomial with a large amount of going up and down on the graph, you're not alone. Most people who regularly use math and polynomial equations can't just look at an equation and get a picture of what it looks like in their head easily. You can pick out little things like the larger the exponent, the larger the increase in the slope, but if you want a good picture of it, you just graph it. Try not to linger too much on what it looks like until you get into calculus.
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Anonymous2005-09-02 17:57
>>5
Assuming you're dealing with just simple polynomials involving only two variables (eg x & y), just think of a polynomial equation as a constraint. OK, not a good choice of words but if I have something like x^2 + y^2 - 1 = 0, only certain pairs of x and y will satisfy the equation. Basically, the equation just describes a relation between certain x points and certain y points.
Obviously, if I give you a random equation with x and y variables, it'll be hard to visualise the shape the points make. So we tend to categorise polynomial equations into lines, ellipsoids, etc. You're expected to memorise these forms so when you see something that looks like a line, you can say, it's a line! And has slope m!
If you're talking about ploynomials with only one variable but with increasing power, then it's a bit different...
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Anonymous2005-09-02 19:11
Interestingly, I've never had to visualize polynomials. It wasn't 'til my first derivatives that I started visualizing things--- It was much easier for me to understand the derivative as a line tangent to the curve at a single point than as the limit of [blah-de-blah I forget the definition].
Polynomials are something that I've always understood in a more algebraic sense, as >>8 describes them "constraints". A sort of fiddling with numbers to state the value of variable X in terms of W, Y, Z, etc.
With respect to the normal slope-intercept form, that's simple rote memorization. y = mx + b. The number m is the slope of the graph because it defines the rate at which Y changes with respect to X. If m=1, then y changes by 1 for every 1 that x changes. If m = 2, then y changes by 2 for every 1 that x changes. If m = 0.5, then y changes by 0.5 for every 1 that x changes.
It also just so happens that the slope is the rate of change of Y with respect to X. (y' - y)/(x' - x) is another way of saying "The difference in y with respect to the difference in x". Now we can decide that we want the difference in x to be 1 by picking the appropriate points on the line, then we can perform the arithmetic and voila! You have the slope!
So m and the slope are really the same thing.
(Note that I'm not providing an actual proof b/c, at that stage, I know I didn't find proofs nearly as convincing as some sort of explanation.)
Good luck, >>5, and don't give up! It really is important to understand your math.
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Anonymous2005-09-03 19:41
Heh. Godel is laughing at you.
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Anonymous2005-09-06 1:42
How can you not love polynomials? How else would you describe your interest in getting 2 scoops as you peruse along a section of 31 flavors of ice cream? And without pistachio ice cream, mankind is nothing.
Alternately, you could say that the polynomial describes a variety of affections you may have, for sports or throngs or thongs or elegance of a sort; the degree and distance to the actual affection are degrees and offsets of polynomial factors. It is easy to produce a model around this which has huge forbidden regions you simply haven't opened up and explored. Some of them are playing for the Knicks in 4' of semisolid frozen lime yogurt with Scrapped Princess sitting on your head and a giant gas grill in front of you, some of them are more like making human-powered mass transport that's cool and dry.
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Anonymous2005-09-06 12:17
I'm a fan of constants, personally
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Anonymous2008-06-29 4:03
bump
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Anonymous2008-07-01 20:41
>"what's the point of it all"
Yeah, who would ever need trigonometry, you dumb motherfucker you.
>>16 GEB?
Gödel, Escher, Bach. A protracted masturbation session by Douglas Hofstadter.
It frightens me to think anyone could find it at all elucidating.
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Anonymous2008-07-03 12:38
Holy fucking shit, almost everyone in this thread is a goddamned idiot.
For fucks sake, polynomials are barely math. Four semesters of calculus? Wow, you're almost there. Even most English majors and shit need to take that much. I will agree that learning logic and how to approach problems is important, but if you're implying that analysis and differential equations doesn't have many practical applications, you're unbelivably fucking stupid and obviously don't know anything about physics, chemistry, statistics, business, economics, or anything actually important.
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Anonymous2008-07-03 14:37
POLYNOMIALS
ARE
MATHEMATICS.
FROM ELEMENTARY SCHOOL TO RESEARCH, THEY'RE EVERYWHERE.
I suggest you begin with Polynomials by E.J. Barbeau for high school problems on polynomials.
Hey guys, I realised my lecturers were full of bullshit when I discovered a proper normal subgroup of the Alternating group on 5 letters. This was validated in a dream I had the night before last, in which a general formula for the solution of degree 5 polynomials presented itself to me. Weird, huh?
Polynomials ARE quite an interesting topic. There are specific behavioural patterns to equations. Personally, I find asymptotes one of the more interesting sub-topics and I'm not quite sure why, after all its just a line a graph approaches but never touches - although in the case of the horizontal asymptote it can cross.
Although I'm a fan of statistics also, but I have to state that algebra is possibly the most intriguing area of mathematics, and polynomials tie in very well, I mean with algebra you can find the roots alpha beta gamma delta etc. of a function or equation by using the coefficients of the equation, the sum of the roots is equal to -b/a
as was stated before at a higher level proofs must be remembered, and things must be proven without aid in exams. It's not too difficult if you like the topic, and algebra and polynomials go hand in hand in higher mathematics.
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Anonymous2008-07-05 0:11
Polynomials are boring as shit until you study abstract algebra and Galois theory, because you can't do anything. Well, okay, they're also pretty badass in complex analysis, too. Either way, at the precalc level, they're gay. Hang in there, it gets better.