Thread: Small things about polynomials

Look at these three expressions:

#1 :
#2 :
#3 :

They are all different representations of the same thing. The second expression is the expanded form of the first. It's called a "quadratic quadrinomial". The third simplifies it by combining the 8x and 3x, but the third is called a "quadratic trinomial". So even though they're both the same thing, they have different names. A few questions...

1. A polynomial is an expression with the 4 arithmetic operations and non-negative exponents. I usually see polynomials as a+b+c+d... etc. where a, b, c, d ... are the terms. Is the first expression above, , a polynomial? I'm not sure, since instead of the usual a+b+c+d..., the most precedent operation is multiplication, not addition.

2. If I expand the first expression, I get the second expression. However, it can be simplified by combining the 8x and 3x, making the third expression. The second is a quadratic quadrinomial, but the third is a quadratic trinomial. Even though they both represent the same thing, they have different names. This leads me to think that, in general, polynomials with different names can (can, not will) essentially be the same thing as long as the highest degree is the same, e.g. a cubic trinomial and a cubic monomial (same degree, different # of terms). But not a cubic quadrinomial and a quadratic trinomial (because different degrees). Is this guessy inference I made true for all polynomials?

Some things you should know

#1 The product of two (or more) polynomials is again a polynomial. Put another way, if a polynomial can be factored it is said to be reducible, the factors themselves being polynomials

#2 Is not a polynomial - you cannot have more than one term of the same degree in a polynomial. In fact it is without real meaning as written

#3 Is a perfectly respectable reducible polynomial of degree 2 which factors as in #1

It is customary to distinguish between polynomial forms and polynomial functions (more later). Two polynomial forms are said to be equal if and only if all their corresponding coefficients are equal.

A polynomial form say cares not a jot what is - it is called an "indeterminate"

A polynomial function cares a great deal. The reason for making the distinction is that, where, say, (p is prime and is the integers modulo p), one has, by a theorem of Fermat (no not that one!) that the distinct forms and specify the same function, namely that which is identically zero

So it looks like I wasted my time - some people have no effing manners

Say what, though - next time this poster asks a question, it won't be ME that tries to be helpful

G.

Guitarist, I'm terribly sorry for not responding. As I was posting to other threads, I forgot to reply, even though I got the email notification. Sorry again, please accept my apology.

Still, thank you very much for your answer. If I may, I'll continue with my questions.

#1 The product of two (or more) polynomials is again a polynomial

Okay. I was confused, since I always see and think of polynomials in a form like term1+term2+term3... As you can see, this one's more of a matter of convention.

Put another way, if a polynomial can be factored it is said to be reducible, the factors themselves being polynomials

So a polynomial multiplied by a polynomial is also a polynomial. And a polynomial divided by a polynomial is also a polynomial. Is this an "if and only if" thing, i.e. can a non-polynomial expression be involved in any of these combinations and result in a polynomial form?

#2 Is not a polynomial - you cannot have more than one term of the same degree in a polynomial.

That makes sense. Hmm... Wouldn't this imply that a polynomial of th degree cannot have more than terms?

Two polynomial forms are said to be equal if and only if all their corresponding coefficients are equal

Seems simple enough, but that for me usually signifies a misunderstanding. Could you provide an example of what you mean exactly?

Originally Posted by halorealm

Guitarist, I'm terribly sorry for not responding

Hi halorealm. Actually I have been reprimanded by a fellow moderator for not allowing you more time to respond. He is right

Also, although he was too polite to say so, I now see my "miff" at not being thanked is a nasty form of vanity on my part. So it is me who should apologize for my strong language.

Okay. I was confused, since I always see and think of polynomials in a form like term1+term2+term3... As you can see, this one's more of a matter of convention.

You are correct, but most errors in thinking about polynomials stem from the fact that one is inclined to forget that a polynomial like is just shorthand for

So a polynomial multiplied by a polynomial is also a polynomial. And a polynomial divided by a polynomial is also a polynomial. Is this an "if and only if" thing, i.e. can a non-polynomial expression be involved in any of these combinations and result in a polynomial form?

Yes, provided we remember the rule above i.e. it may "look" like say a scalar

Wouldn't this imply that a polynomial of th degree cannot have more than terms?

Yup, nor less, from the above rule

Could you provide an example of what you mean exactly?{by the equality of polynomial forms - my insertion}

where we have no interest in whether or even if this equality has any meaning in the more general sense

Originally Posted by Guitarist

not allowing you more time to respond

It's fine. I'll admit my part on the fault for carelessly forgetting your response. No harm done on your part as I see it.

As for your answers, most of them I've learned but I'm not sure about a few things.

Yes, provided we remember the rule above i.e. it may "look" like say a scalar

Forgive me if this is a stupid of me. I'm only aware of scalars in the context of vectors and matrices. I'm confused as to how this relates in the case of polynomials.

You are correct, but most errors in thinking about polynomials stem from the fact that one is inclined to forget that a polynomial like is just shorthand for

True. But again, the most precedent operation is still addition. My form is mainly multiplication, as opposed to what I always see "this+that+whatever". Considering people often have a looser interpretation of polynomials than the actual definition makes, I'm just cautious to see if an actual product of polynomials, regardless of it being equal to a polynomial, is actually a polynomial itself written out like that. Remember that though #2 was equal to #3, it was not considered a polynomial in that form.