Is This Function a Polynomial?

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In summary, the conversation discusses whether a given function is a polynomial and its properties. The function $2x^{-3}-2x^{2}+3$ is not a polynomial since it has a negative exponent. Polynomials must have non-negative integer exponents. It also mentions that polynomials are continuous, differentiable, well-behaved, and have no asymptotes. The conversation provides a resource with examples of polynomials for further understanding.
  • #1
karush
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$2x^{-3}-2x^{2}+3$

Decide whether the function is a polynomial function, If so, states its degree, type, and leading coefficient.

well, I presume it is a polynomial since it the sum of powers in the variable x. Its degree is 2 since that is highest power, but I don't know its type, since it is not quadratic or cubic. and has a vertical asymptote. and of course the leading coefficient is 2

couldn't find the answer in the book?
 
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  • #2
Polynomials cannot have negative powers of $x$, so it's not a polynomial. Polynomials are continuous everywhere, differentiable everywhere, are well-behaved and have no asymptotes.

See the Wikipedia article, it says "non-negative integer exponents" :)
 
  • #3
One of the criteria for an expression to be a polynomial is that the exponents must be non-negative integers. So given that piece of information, what do you say the answer is?

Here's a site that goes over some examples.
 
  • #4
this is a overleaf project I am working lots of pre calc problems
140 pdf 100 pages
 

Related to Is This Function a Polynomial?

1. What is a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. It can have one or more terms, and the highest power of the variable is always a non-negative integer.

2. How do you classify polynomial types?

Polynomials are classified based on the number of terms they have. A polynomial with one term is called a monomial, two terms is called a binomial, three terms is called a trinomial, and any polynomial with more than three terms is called a polynomial with more than three terms is called a polynomial.

3. What is a degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the expression. For example, the degree of the polynomial 3x^2 + 2x + 5 is 2, as the highest power of x is 2.

4. What is a linear polynomial?

A linear polynomial is a polynomial with a degree of 1, meaning it has one term with a variable raised to the first power. It can be written in the form ax + b, where a and b are constants.

5. How do you simplify polynomial expressions?

To simplify a polynomial expression, you combine like terms by adding or subtracting the coefficients. You can also use the distributive property to factor out common factors and simplify the expression. Additionally, you can use polynomial long division or synthetic division to divide polynomials and simplify the expression.

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