Expand expands out products and powers, writing the polynomial as a simple sum of terms:
Factor performs complete factoring of the polynomial:
FactorTerms pulls out the overall numerical factor from t:
There are several ways to write any polynomial. The functions Expand, FactorTerms, and Factor give three common ways. Expand writes a polynomial as a simple sum of terms, with all products expanded out. FactorTerms pulls out common factors from each term. Factor does complete factoring, writing the polynomial as a product of terms, each of as low degree as possible.
When you have a polynomial in more than one variable, you can put the polynomial in different forms by essentially choosing different variables to be "dominant". Collect[poly,x] takes a polynomial in several variables and rewrites it as a sum of terms containing different powers of the "dominant variable" x.
Here is a polynomial in two variables:
Collect reorganizes the polynomial so that x is the "dominant variable":
If you specify a list of variables, Collect will effectively write the expression as a polynomial in these variables:
The Wolfram System does not automatically expand out expressions of the form Null except when Null is an integer. In general it is only correct to do this expansion if Null and Null are positive reals. Nevertheless, the function PowerExpand does the expansion, effectively assuming that Null and Null are indeed positive reals.
The Wolfram System does not automatically expand out this expression:
PowerExpand does the expansion, effectively assuming that x and y are positive reals: