How Do You Differentiate Between Polynomial and Rational Algebraic Functions?

In summary, the conversation discusses the differences between polynomials and rational algebraic functions. The main difference is that a polynomial is a linear combination of non-negative powers of x, while a rational algebraic function is a fraction with polynomials in the numerator and denominator. Examples 1 and 2 are polynomials while examples 3 and 4 are rational algebraic functions. The conversation also touches on the definition of a polynomial and a rational algebraic function, as well as a tip for properly applying exponents in mathematical expressions.
  • #1
JasonRox
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How can you tell the two apart?

Here are some examples in the book:

1. [tex]3x^3 + 2x + 1[/tex]

2. [tex]3x^2 + (x + 1)^1/2[/tex]

3. [tex]\frac{2x + 3}{x^2 + 1}[/tex]

4. [tex](\frac{x}{x + 1})^X[/tex]
 
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  • #2
You just need to distribute.

#2 distributed is:
[itex] 3x^2 + 1/2x + 1/2 [/itex]

and #4 distributed is:
[itex] \frac{x^2}{x+1} [/itex]

After distributing all four are very obviously different
 
  • #3
After relooking at your post I realized that you may not be comparing equations but instead classifying them.

A polynomial can be express as [itex] ax^n + bx^(n-1) + cx^(n-2) + dx^(n-3) … +c [/itex]

I’d have to look up the definition for a “rational algebraic expression” but going from memory it is any expression that has only algebraic terms?
 
  • #4
the polynomial 's power must be postive interger . that's the difference
 
  • #5
A Polynomial (in x) is a linear combination of non-negative powers of x.

A rational algebraic function is just a fraction N(x)/D(x) where N and D are both polynomials.

In your examples 1. and 2. are polynomials while 3. and 4. are rational algebraic functions.
 
  • #6
uart said:
A Polynomial (in x) is a linear combination of non-negative powers of x.

A rational algebraic function is just a fraction N(x)/D(x) where N and D are both polynomials.

In your examples 1. and 2. are polynomials while 3. and 4. are rational algebraic functions.


oh? i though the power of polynomial must be interger, i go check
 
  • #7
expscv said:
oh? i though the power of polynomial must be interger, i go check

No need to check you're correct. It was just a slip, I meant to say non-negative integer but only type non-negative. :eek:
 
  • #8
Thanks, guys.
 
  • #9
JonF said:
A polynomial can be express as [itex] ax^n + bx^(n-1) + cx^(n-2) + dx^(n-3) ? +c [/itex]
JonF: use curly brackets to apply something (in this case ^) to an expression.
[itex] ax^n + bx^{n-1} + cx^{n-2} + dx^{n-3}...+c [/itex]
 
  • #10
Why would the constant term be equal to the coefficient in front of x^(n - 2)? ;)
 
  • #11
argh, thank you
 

FAQ: How Do You Differentiate Between Polynomial and Rational Algebraic Functions?

What are polynomial functions?

Polynomial functions are algebraic functions that involve only non-negative integer powers of the independent variable x. They can be written in the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where an, an-1, ..., a1, a0 are constants and n is a non-negative integer.

What are rational algebraic functions?

Rational algebraic functions are expressions that involve both polynomial functions and rational expressions. They can be written in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions and Q(x) is not equal to 0.

How do I simplify polynomial and rational algebraic functions?

To simplify polynomial and rational algebraic functions, you can use techniques such as factoring, combining like terms, and finding common factors. It is important to remember the rules of operations for polynomials, such as the distributive property and the rules for exponents.

How do I solve equations involving polynomial and rational algebraic functions?

To solve equations involving polynomial and rational algebraic functions, you can use algebraic techniques such as isolating the variable and using inverse operations. It is also helpful to have a good understanding of the properties of equality and how to solve equations with fractions.

How can I apply polynomial and rational algebraic functions in real-life situations?

Polynomial and rational algebraic functions can be applied in various real-life situations, such as in economics, physics, and engineering. They can be used to model and solve problems involving rates of change, optimization, and other mathematical concepts. For example, in economics, polynomial functions can be used to model profit and cost functions, while rational functions can be used to model supply and demand functions.

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