Polynomial Functions

[mathdad]

Determine which functions are polynomial functions.

1. f(x) = 5x^2 + 4x^4

2. h(x) = 3 - (x/2)

3. F(x) = (x^2 - 5)/(x^3)


NOTE: I seek solution steps only. I want to try this on my own (following your steps) before posting my work here.

 

[MarkFL]

A polynomial function may be written in the form:

[MATH]f(x)=\sum_{k=0}^n\left(a_kx^k\right)[/MATH]
In other words, it will be the sum of one or more terms each having a coefficient times the independent variable raised to a non-negative integer exponent. You will find that only one of the given choices does not meet the criteria.

 

[mathdad]

A polynomial function may be written in the form:

[MATH]f(x)=\sum_{k=0}^n\left(a_kx^k\right)[/MATH]
In other words, it will be the sum of one or more terms each having a coefficient times the independent variable raised to a non-negative integer exponent. You will find that only one of the given choices does not meet the criteria.

I need to know what defines a polynomial function. Can you provide the steps?

 

[Dr.Peterson]

I'd say that the first "step" to answering these questions is to look up the definition of polynomial. Show us the definition you were given, and we can help you see how to apply it to specific expressions.

If your book (or other source) doesn't have a definition, then it is deficient for your purposes.

But what MarkFL gave you is a definition. Are you asking for a definition because you don't understand that one, or is there some other reason that answer wasn't sufficient? I'm not sure what you need from us.

 

[pka]

I need to know what defines a polynomial function. Can you provide the steps?

Determine which functions are polynomial functions.
1. f(x) = 5x^2 + 4x^4
2. h(x) = 3 - (x/2)
3. F(x) = (x^2 - 5)/(x^3)

Once again, MarkFl has given you the answer. \(\displaystyle \sum\limits_{k = 0}^n {{a_k}{x^k}} \) is a polynomial provided each \(\displaystyle a_k\) is a number(not a function, i.e not a variable).
Let's take each of the three given functions.
1) \(\displaystyle f(x)=5x^2+4x^4=\sum\limits_{k = 0}^4 {{a_k}{x^k}} \) so that \(\displaystyle a_0=0,~a_1=0,~a_2=5,~a_3=0,~\&~a_4=4\)
By definition is that that a polynomial ? Why or why not?

2) \(\displaystyle h(x)=3x^0+\frac{-1}{2}x=\sum\limits_{k = 0}^1 {{a_k}{x^k}} \) so that \(\displaystyle a_0=?,~a_1=?\)
By definition is that that a polynomial ? Why or why not?

3) \(\displaystyle F(x)=\frac{x^2-5}{x^3}=x^{-1}-5x^{-3}\)
By definition is that that a polynomial ? Why or why not?

 

[Otis]

I need to know what defines a polynomial function. Can you provide the steps?

Step 1: Google keywords polynomial math definition

Step 2: Read the definitions at sites like the following.

https://www.purplemath.com/modules/polydefs.htm

Polynomials intro (video) | Khan Academy

Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Introduction to polynomials. This video covers common terminology like terms, degree, standard form, monomial, binomial and trinomial.

www.khanacademy.org

What is a Polynomial? | Virtual Nerd

Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique...

virtualnerd.com

\(\;\)

 

[mathdad]

I'd say that the first "step" to answering these questions is to look up the definition of polynomial. Show us the definition you were given, and we can help you see how to apply it to specific expressions.

If your book (or other source) doesn't have a definition, then it is deficient for your purposes.

But what MarkFL gave you is a definition. Are you asking for a definition because you don't understand that one, or is there some other reason that answer wasn't sufficient? I'm not sure what you need from us.

I will go back into the textbook to define what a polynomial function is as expressed by Sullivan.

 

[mathdad]

Once again, MarkFl has given you the answer. \(\displaystyle \sum\limits_{k = 0}^n {{a_k}{x^k}} \) is a polynomial provided each \(\displaystyle a_k\) is a number(not a function, i.e not a variable).
Let's take each of the three given functions.
1) \(\displaystyle f(x)=5x^2+4x^4=\sum\limits_{k = 0}^4 {{a_k}{x^k}} \) so that \(\displaystyle a_0=0,~a_1=0,~a_2=5,~a_3=0,~\&~a_4=4\)
By definition is that that a polynomial ? Why or why not?

2) \(\displaystyle h(x)=3x^0+\frac{-1}{2}x=\sum\limits_{k = 0}^1 {{a_k}{x^k}} \) so that \(\displaystyle a_0=?,~a_1=?\)
By definition is that that a polynomial ? Why or why not?

3) \(\displaystyle F(x)=\frac{x^2-5}{x^3}=x^{-1}-5x^{-3}\)
By definition is that that a polynomial ? Why or why not?

I totally get it. Thanks.

 

[mathdad]

Step 1: Google keywords polynomial math definition

Step 2: Read the definitions at sites like the following.

https://www.purplemath.com/modules/polydefs.htm

Polynomials intro (video) | Khan Academy

Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Introduction to polynomials. This video covers common terminology like terms, degree, standard form, monomial, binomial and trinomial.

www.khanacademy.org

What is a Polynomial? | Virtual Nerd

Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique...

virtualnerd.com

\(\;\)

Helpful links for my study time.

 

[mathdad]

A polynomial function may be written in the form:

[MATH]f(x)=\sum_{k=0}^n\left(a_kx^k\right)[/MATH]
In other words, it will be the sum of one or more terms each having a coefficient times the independent variable raised to a non-negative integer exponent. You will find that only one of the given choices does not meet the criteria.

Determine which functions are polynomial functions.

1. f(x) = 5x^2 + 4x^4

Let PF = polynomial function

Question 1 is a PF.

2. h(x) = 3 - (x/2)

This is a PF.

3. F(x) = (x^2 - 5)/(x^3)

This is not a PF.

Book's Reason:

"The polynomial in the denominator is of positive degree."

What does the book mean here?

4. g(x) = sqrt{x}

This is not a PF.

Book's Reason:

We say that g(x) = sqrt{x} is the same as x^(1/2), meaning that x is raised to the 1/2 power, which is "not a nonnegative integer."

Note: I paraphrased part of the book's reason for question 4. What does the book mean by
"not a nonnegative integer"?

 

[Dr.Peterson]

Determine which functions are polynomial functions.

1. f(x) = 5x^2 + 4x^4

Let PF = polynomial function

Question 1 is a PF.

2. h(x) = 3 - (x/2)

This is a PF.

3. F(x) = (x^2 - 5)/(x^3)

This is not a PF.

Book's Reason:

"The polynomial in the denominator is of positive degree."

What does the book mean here?

Again, it might be helpful to see the exact wording of the book's definition; maybe it mentioned denominators, or maybe there was a subsequent comment that this refers back to. This is a rather odd way to say it.

My guess as to what they mean is that a polynomial can involve division, but only with mere numbers in the denominator -- that is, the coefficient can be a fraction, but you can't be dividing by anything with a variable in it. A polynomial of positive degree means that the degree (which must be a non-negative integer) is not zero -- that is, the "polynomial" is not just a constant.

4. g(x) = sqrt{x}

This is not a PF.

Book's Reason:

We say that g(x) = sqrt{x} is the same as x^(1/2), meaning that x is raised to the 1/2 power, which is "not a nonnegative integer."

Note: I paraphrased part of the book's reason for question 4. What does the book mean by
"not a nonnegative integer"?

A non-negative integer is any of 0, 1, 2, 3, ... . Since 1/2 is not one of these, it is not a non-negative integer. And since the exponent is not a non-negative integer, this is not a valid term for a polynomial.

Are you familiar with the fact that square roots are equivalent to 1/2 powers, as stated in the answer?

 

[mathdad]

Again, it might be helpful to see the exact wording of the book's definition; maybe it mentioned denominators, or maybe there was a subsequent comment that this refers back to. This is a rather odd way to say it.

My guess as to what they mean is that a polynomial can involve division, but only with mere numbers in the denominator -- that is, the coefficient can be a fraction, but you can't be dividing by anything with a variable in it. A polynomial of positive degree means that the degree (which must be a non-negative integer) is not zero -- that is, the "polynomial" is not just a constant.


A non-negative integer is any of 0, 1, 2, 3, ... . Since 1/2 is not one of these, it is not a non-negative integer. And since the exponent is not a non-negative integer, this is not a valid term for a polynomial.

Are you familiar with the fact that square roots are equivalent to 1/2 powers, as stated in the answer?

Yes, I know that square roots are equivalent to 1/2 powers. Good notes here.

 

[Steven G]

A polynomial is just the sum/difference of monomials. So it is best to define a monomial.
A monomial is in the form of axⁿ where a is any real number and n is a whole number. Recall the whole numbers is W = {0,1,2,3,4,5....}