How Do You Differentiate Between Polynomial and Rational Algebraic Functions?

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Discussion Overview

The discussion focuses on differentiating between polynomial functions and rational algebraic functions, exploring definitions, examples, and characteristics of each type. Participants examine specific examples to clarify their classifications and underlying principles.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that a polynomial can be expressed as a sum of terms with non-negative integer powers of x.
  • Others propose that a rational algebraic function is defined as a fraction where both the numerator and denominator are polynomials.
  • One participant mentions the need to distribute terms to clarify the differences between the examples provided.
  • Another participant questions the definition of a rational algebraic expression, indicating uncertainty about its characteristics.
  • There is a correction regarding the requirement that polynomial powers must be non-negative integers, with some participants acknowledging this clarification.
  • Participants express confusion about the relationship between constant terms and coefficients in polynomial expressions.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of polynomials and rational algebraic functions, but there are some uncertainties and clarifications regarding the specifics of polynomial powers and the classification of examples. The discussion contains multiple viewpoints and some unresolved questions.

Contextual Notes

Some participants express uncertainty about definitions and characteristics, particularly regarding rational algebraic expressions and the nature of polynomial powers. There are also unresolved mathematical expressions and notation issues raised in the discussion.

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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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
 
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?
 
the polynomial 's power must be positive interger . that's the difference
 
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.
 
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
 
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. :o
 
Thanks, guys.
 
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
 

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