Definition of a polynomial? and degree? integral and ration

[Kilo Vectors]

Hello

What is the standard definition of a polynomial? according to the book I am using a polynomial is an algebraic expression which is integral and rational for all the terms.

It gives no definition of integral or rational seperately, but I think integral means that the variables are to powers of integers and not fractions such that a/b and b=/= 1 basically non whole numbers

What does rational mean?

And the degree of a polynomial? Schaumms outline says it is the sum of all the exponents of variables. But my school taught that its the value of the variable with greatest exponent.

Dont know if thread belongs here. Sorry if wrong section.

 

[fresh_42]

Hello

What is the standard definition of a polynomial? according to the book I am using a polynomial is an algebraic expression which is integral and rational for all the terms.

It gives no definition of integral or rational seperately, but I think integral means that the variables are to powers of integers and not fractions such that a/b and b=/= 1 basically non whole numbers

What does rational mean?

And the degree of a polynomial? Schaumms outline says it is the sum of all the exponents of variables. But my school taught that its the value of the variable with greatest exponent.

Dont know if thread belongs here. Sorry if wrong section.

A polynomial ##p## in one variable ##x## over a set ##R## is a function ##p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0## where all ##a_n, ... , a_0 \in R##.
##R## is usually a special form of set, called a ring or a field. ℤ is the ring of integers, ℚ the field of rational numbers, and ℝ the real numbers.
If ℤ is meant you can say polynomial over the integers (not integral) and if ℚ is meant you can say polynomial over the rationals or polynomial with rational coefficients. Coefficients are the ##a_i##.
And you are right, ##n## is the degree of the polynomial. However, if you have more than one variable, say ##x## and ##y## and the highest term is, e.g. ##x^3 y^5## then the degree of the polynomial is ##8##.

 

[SteamKing]

To be clear, a "rational" number is one which can be expressed as the ratio of two integers p and q, such that the rational number r = p / q, where q ≠ 0.

Since q can equal 1, then the set of integers is a subset of the set of rational numbers.

https://en.wikipedia.org/wiki/Rational_number

 

[Kilo Vectors]

To be clear, a "rational" number is one which can be expressed as the ratio of two integers p and q, such that the rational number r = p / q, where q ≠ 0.

Since q can equal 1, then the set of integers is a subset of the set of rational numbers.

https://en.wikipedia.org/wiki/Rational_number

Hello Mr Steam, well first off thank you for answering. I know what rational numbers are but I am not sure that is what the book means when it refers to them. In fact it says all terms are rational and integral, so I have no idea what a rational variable is? I think Mr Fresh post solved my question though.

 

[Kilo Vectors]

A polynomial ##p## in one variable ##x## over a set ##R## is a function ##p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0## where all ##a_n, ... , a_0 \in R##.
##R## is usually a special form of set, called a ring or a field. ℤ is the ring of integers, ℚ the field of rational numbers, and ℝ the real numbers.
If ℤ is meant you can say polynomial over the integers (not integral) and if ℚ is meant you can say polynomial over the rationals or polynomial with rational coefficients. Coefficients are the ##a_i##.
And you are right, ##n## is the degree of the polynomial. However, if you have more than one variable, say ##x## and ##y## and the highest term is, e.g. ##x^3 y^5## then the degree of the polynomial is ##8##.

Hi Mr Fresh, first of thank you so much for taking the time to answer so elaborately. I very much appreciate it..Ok I think I understand.

 

[SteamKing]

Hello Mr Steam, well first off thank you for answering. I know what rational numbers are but I am not sure that is what the book means when it refers to them. In fact it says all terms are rational and integral, so I have no idea what a rational variable is?

I don't know what that means, either. Can you provide a direct quote from your textbook, without paraphrasing it? Better yet, if you could upload a scan of the passage which defines the term "polynomial" from your text.

 

[Kilo Vectors]

I don't know what that means, either. Can you provide a direct quote from your textbook, without paraphrasing it? Better yet, if you could upload a scan of the passage which defines the term "polynomial" from your text.

Yes I will. Just wait one or two minutes while I arrange this

 

[Kilo Vectors]

 

[Kilo Vectors]

I don't know what that means, either. Can you provide a direct quote from your textbook, without paraphrasing it? Better yet, if you could upload a scan of the passage which defines the term "polynomial" from your text.

here I posted it above thank you. My knowledge is very poor in maths so I am just starting from the basics.

 

[Kilo Vectors]

A polynomial ##p## in one variable ##x## over a set ##R## is a function ##p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0## where all ##a_n, ... , a_0 \in R##.
##R## is usually a special form of set, called a ring or a field. ℤ is the ring of integers, ℚ the field of rational numbers, and ℝ the real numbers.
If ℤ is meant you can say polynomial over the integers (not integral) and if ℚ is meant you can say polynomial over the rationals or polynomial with rational coefficients. Coefficients are the ##a_i##.
And you are right, ##n## is the degree of the polynomial. However, if you have more than one variable, say ##x## and ##y## and the highest term is, e.g. ##x^3 y^5## then the degree of the polynomial is ##8##.

here I posted it above thank you. My knowledge is very poor in maths so I am just starting from the basics.

 

[SteamKing]

Thanks for making an image of this passage.

I see what the author has tried to do here. In the middle of the passage, he states, "A term is integral and rational in certain literals (letters which represent numbers) if the term consists of
(a) positive integer powers of the variables multiplied by a factor not containing any variable, or
(b) no variable at all."

A better statement would be that the variables in the terms of a polynomial all have integer powers, while the multiplicative coefficients are all rational numbers. Zero is an integer power, therefore, a constant term which is not multiplied by any variable can also be present in a polynomial.

 

[Kilo Vectors]

Thanks for making an image of this passage.

I see what the author has tried to do here. In the middle of the passage, he states, "A term is integral and rational in certain literals (letters which represent numbers) if the term consists of
(a) positive integer powers of the variables multiplied by a factor not containing any variable, or
(b) no variable at all."

A better statement would be that the variables in the terms of a polynomial all have integer powers, while the multiplicative coefficients are all rational numbers. Zero is an integer power, therefore, a constant term which is not multiplied by any variable can also be present in a polynomial.

Ah now it makes sense..very well put. Thank you so much.