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I am new to Regression and R. I know that polynomial functions are used when a regression model does not fit data (underfitting), but I want to know which degree of polynomial should be used? I also want to know that if a regression model uses multiple variables (e.g., y ~ x1+x2+x3+x4) then do I need a polynomial function in this model?

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That depends on the problem under consideration. A polynomial regression always yields a better fit than an affine regression but it could make no sense from the interpretation perspective. – Stéphane Laurent Sep 4 '12 at 13:27
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By presupposing a polynomial solution, this question implicitly rules out procedures that might be simpler or more effective. I would like to suggest that you begin your research by reviewing any appealing threads related to model selection. You may find many of these informative and stimulating. Then consider returning with a more specific question motivated by that information. – whuber♦ Sep 4 '12 at 15:46
Did you checked for linearity!? check for other models for non linear data. or you can always plot the data and try to build a polynomial model that as the same look...then just shape it (fit it)... – user13840 Sep 4 '12 at 16:55

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Use polynomial terms if that seems appropriate. As Stephane mentions you can always get better fits with more complicated models but you could be overfitting to the noise. The way to determine the order of the polynomial is to introduce terms of various degrees and test to see which ones if any have their regression coefficient statistically significantly different from 0.

When several variables are involved the appropriateness of polynomial terms is no different than if only one covariate is used. You include polynomial terms only if that seems to be appropriate to fitting a model to the data.

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I think this question has to do with exactly how to choose the order of the polynomial. The right answer may be "look at a plot of the data" but I'm having trouble seeing how to use this answer, particularly the second paragraph. The polynomial terms and cross products are most likely going to be collinear so how exactly do you suggest choosing variables? Because of the collinearity, it seems that the order of testing could have an impact on the final result, which isn't good. Do you have a way around that? – Macro Sep 4 '12 at 14:45
@Macro I think you could do this sequentially starting with the quadratic term first and then moving up to degree 3 and 4 etc. I wouldn't look at interaction terms unless they are expected. – Michael Chernick Sep 4 '12 at 14:53
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Using what test/criteria? If you're doing this with multiple different variables, how exactly would you do that - incrementally increase the order of one variable and, once you're done with that, move on to the next variable? Why exactly does that make sense? Why wouldn't you just start with a high order and pair your way down from there? What does "Use polynomial terms if that seems appropriate" mean in your opening sentence? These are just a few things that need clarification for this answer to be useful to a layman (or a statistician, actually). – Macro Sep 4 '12 at 15:01
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Also note that a polynomial expansion of a multi-variable function necessarily includes cross products between the variables so it doesn't necessarily make sense to a priori exclude these cross products from consideration any more than it makes sense to exclude certain variables a priori. To be fair, I think this question is too vague in its present form to answer (the biggest question is why is the OP including polynomial terms???) but since you chose to answer it, I'm bringing these issues to you. – Macro Sep 4 '12 at 15:05
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I agree the question is too vague. There's no general answer to such a question, it is specific to the problem/data under consideration. – Stéphane Laurent Sep 4 '12 at 15:27

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