Super basic polynomial and exponent definition help

In summary, the conversation is about understanding the form of a polynomial and how to recognize and apply it. The polynomial in question has the general form ax5 + bx3 + cx0, and the coefficients are 3, 7, and -5. The last term is -5, not 5, and it is considered a constant or absolute term. The concept of coefficients and exponents can be confusing, but it is important to understand in order to work with polynomials. Some resources like Khan Academy may be helpful in understanding polynomials.
  • #1
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Please bare with me. Most of you know I actually don't have a great math background. In any case I'm going way back and filling in some very basic math that I have long forgot. I have some questions about terms in a polynomial.

Here is an example

$$3x^5+7x^3-5$$

1. From my book 3 and 7 are clearly coefficients, but I am unsure if 5 is also one? My understanding is that coefficients are in front of a variable.

2. Would the last term be 5 or -5? Both? Since it could be plus -5 for the same result? If you were to list that term how would you do it?

3. Even though there is no exponent listed on the 5, is ##5^1## implied and identical? Certainly it would not be ##5^0## because that always equals 1 right?

Thanks!
 
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  • #2
The polynomial you have is of the general form ax5 + bx3 + cx0, where a = 3, b =7 and c = -5.
 
  • #3
kuruman said:
The polynomial you have is of the general form ax5 + bx3 + cx0, where a = 3, b =7 and c = -5.
ok so there is an implied variable behind the 5. I guess I get c = -5 but I am unsure why the exponent is 0? From what I read any term with an exponent of 0 is 1.

Furthermore why is subtraction actually a thing. This is example it's confusing and should just be plus -5. That makes it so clear instead of having to do a mental check.
 
  • #4
Is this homework?
 
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  • #5
Greg Bernhardt said:
Please bare with me. Most of you know I actually don't have a great math background. In any case I'm going way back and filling in some very basic math that I have long forgot. I have some questions about terms in a polynomial.

Here is an example

$$3x^5+7x^3-5$$

1. From my book 3 and 7 are clearly coefficients, but I am unsure if 5 is also one? My understanding is that coefficients are in front of a variable.
Basically yes, the "co" means "at" so a variable is needed, but it's more convenient to write ##3x^5+7x^3-5 \triangleq (3,0,7,0,0,-5)## and call all of them by the same term. If you like then it could be considered as ##- 5= -5 \cdot 1 = -5\cdot x^0## and then there is the variable again.
2. Would the last term be 5 or -5? Both? Since it could be + -5 for the same result? If you were to list that term how would you do it?
##-5##, see ##(3,0,7,0,0,-5)## above.
3. Even though there is no exponent listed on the 5, is ##5^1## implied and identical? Certainly it would not be ##5^0## because that always equals 1 right?
Yes, but ##x^0## is the solution here. It is called absolute term or absolute coefficient, and cannot be decoupled from the polynomial, neither mathematically nor by name. And it is really needed, as it guarantees the embedding of the numbers in the ring of polynomials ##\mathbb{Q} \hookrightarrow \mathbb{Q}[x]##. Numbers are also polynomials ##p(x)=1\,.##
 
  • #6
The -5 is just a convenience just like we use contractions in everyday English. You just implicitly think + -5

also the polynomial has some missing terms for ##x^4, x^2## and ##x^1## with zero coefficients.
 
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  • #7
fresh_42 said:
Basically yes, the "co" means "at" so a variable is needed, but it's more convenient to write ##3x^5+7x^3-5 \triangleq (3,0,7,0,0,-5)## and call all of them by the same term. If you like then it could be considered as ##- 5= -5 \cdot 1 = -5\cdot x^0## and then there is the variable again.
##-5##, see ##(3,0,7,0,0,-5)## above.
Yes, but ##x^0## is the solution here. It is called absolute term or absolute coefficient, and cannot be decoupled from the polynomial, neither mathematically nor by name. And it is really needed, as it guarantees the embedding of the numbers in the ring of polynomials ##\mathbb{Q} \hookrightarrow \mathbb{Q}[x]##. Numbers are also polynomials ##p(x)=1\,.##

Fresh I'm at a 1st grade level here! :biggrin:
But I get some of it and it seems that the key to my questions really is that one needs to understand the form which @kuruman posted in #2. Perhaps have jumped ahead somewhere.
 
  • #9
Greg Bernhardt said:
From what I read any term with an exponent of 0 is 1.
That is absolutely correct. The constant in a polynomial can be viewed as the coefficient of the zeroth power. This allows writing the most general polynomial as
$$P=\sum_{n=0}^{N}a_nx^n$$
The coefficients ##a_n## can be any number, positive negative or zero. The highest exponent, N, denotes the order of the polynomial.

On edit: Corrected in light of comments below.
 
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  • #10
fresh_42 said:
If you like then it could be considered as ##- 5= -5 \cdot 1 = -5\cdot x^0## and then there is the variable again.
This now makes sense, thanks! :smile:

Just to be clear, one has to learn the form and be able to recognize to apply the form when seeing that polynomial? Because inherently it doesn't seem obvious that I should know to add an ##x^0## after the -5.
 
  • #12
Aha doing your kids schoolwork. There ought to be a law against that... also Khan Academy should have some good videos on polynomials.
 
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  • #13
jedishrfu said:
Aha doing your kids schoolwork. There ought to be a law against that... also Khan Academy should have some good videos on polynomials.
Why to be satisfied with second best? :cool:

Just thought today: Maybe we don't solve your problems for you, but we definitely improve your understanding!
 
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  • #14
kuruman said:
That is absolutely correct. The constant in a polynomial can be viewed as the coefficient of the zeroth power. This allows writing the most general polynomial as
$$f(x)=\sum_{n=0}^{\infty}a_nx^n$$
The coefficients ##a_n## can be any number, positive negative or zero.

ummm, someone may correct me, but I don't think this is right.

My understanding is that all single variable polynomials are nilpotent with respect to the derivatives operator -- or perhaps more to the point: all polynomials are of finite degree. What you've written is a power series or perhaps generating function, which isn't a polynomial per se.
 
  • #15
StoneTemplePython said:
ummm, someone may correct me, but I don't think this is right.

My understanding is that all single variable polynomials are nilpotent with respect to the derivatives operator -- or perhaps more to the point: all polynomials are of finite degree. What you've written is a power series or perhaps generating function, which isn't a polynomial per se.

The point is that such a polynomial has only a finite amount of coefficients that are non zero, so that we really can write it that way. That we write this as an infinite sum, is just notation:

In my algebra course, we rigorously defined polynomials as sequences of elements. For example, the sequence

##(1,2,3,0,0,0, \dots)## denotes the polynomial ##1 + 2x + 3x^2##. And the latter is just the way we represent the sequence of polynomials. Also, addition and multiplication are defined on those sequences.

From this, you can see that you are actually confusing polynomials and polynomial functions, but this is ok, because we can identify the two things with each other, but they are not formally the same things.
 
  • #16
StoneTemplePython said:
ummm, someone may correct me, but I don't think this is right.
You are right. There should have been the additional information almost all ##a_n=0##, for otherwise it's a series and not a polynomial.
 
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What is a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The variables can only have non-negative integer exponents.

What is an exponent?

An exponent is a number that represents the power to which a base number is raised. It is written as a small number to the right and above the base number. For example, in 23, 3 is the exponent.

How do you simplify a polynomial?

To simplify a polynomial, you combine like terms by adding or subtracting coefficients that have the same variables and exponents. You can also use the distributive property to remove parentheses and combine like terms.

What is the degree of a polynomial?

The degree of a polynomial is the highest exponent in the expression. For example, the degree of 3x2 + 2x + 7 is 2.

What is the difference between a monomial, binomial, and trinomial?

A monomial is a polynomial with one term, such as 5x. A binomial has two terms, such as 2x + 3. A trinomial has three terms, such as 3x2 + 4x + 5.

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