Addition
Mathematicians love arithmetic. No matter how fancy the objects are, it seems like we always want to find ways to add, subtract, multiply and divide them. Polynomials are no exception. In this part of the course, we're going to look at how we do all of those basic operations on polynomials. The Rules . . .
I think the best way to approach this material is by looking at some specific examples. 

Quick Definition  "Like Terms"
This is a phrase that gets thrown around a lot when people are talking about polynomials and it often doesn't get carefully defined. When we refer to like terms, we mean two terms that have the same variable parts. So, for example, in the expression x^{2} + 3x^{2}  2x^{3}, x^{2} and 3x^{2} are 'like terms' because they both have exactly the same variable part, i.e. the x^{2}. On the other hand, x^{2} and 2x aren't like terms because their variable parts are different  one has a two in the exponent where the other has a three.
Example 1
Simplify 2x^{2} + 3x^{2}.
In this case, the direction to "simplify" is asking us to make the polynomial "smaller" by adding together any "like terms". Since the two terms in the expression both have exactly the same variable parts, x^{2}, we can combine them by adding their variable parts:
2x^{2} + 3x^{2} = (2 + 3)x^{2} = 5x^{2}
Example 2
Simplify x^{3}  3x^{2}.
This one can't be simplified any further. Notice that the exponents on the variables are different  two versus three. Because they aren't the same, those terms can't be added together.
Example 3  The Grouping Method
Simplify 2x^{3} + 3y^{3}  3x^{3}.
The first thing we need to do is rearrange the terms (being very careful of negative signs) so that all the like terms are together. In this case, that means moving the x^{3} terms:
2x^{3}  3x^{3} + 3y^{3}
The two x terms can be combined together to make (2  3)x^{3} = x^{3}. There's nothing we can do with the y^{3} term. It does have the same exponent as the other two terms but, because the variables are different  x versus y, we can't combine it with the other two terms. That makes the simplified version:
2x^{3}  3x^{3} + 3y^{3} = (2  3)x^{3} + 3y^{3} = x^{3} + 3y^{3}
Example 3  The Table Method
Add the polynomials 3x^{4} + 2x^{3} + 4x  5 and 2x^{3}  4x^{2} + 4x + 2.
When you've got two large polynomials like these two work with, a strategy that's often helpful is to write the two polynomials, one over the other, with their like terms lined up. Then you can just add down the columns. With these two polynomials it would like looks this:
3x^{4}  + 2x^{3}  + 4x   5  
2x^{3}   4x^{2}  + 4x  + 2 
Notice how some of the terms, e.g. the 4x^{2} one in the second polynomial, don't have a match in the other polynomial. That's okay. That term will just get carried down to the final answer, unchanged. You can think of it as adding a zero to that term. (I'll even write it that way in the next part of the solution.)
Now to add the polynomials together, we just add down the columns:
3x^{4}  + 2x^{3}  + 4x   5  
2x^{3}   4x^{2}  + 4x  + 2  
3x^{4}  + (2  2)x^{3}   4x^{2}  + (4 + 4)x   5 + 2 
3x^{4}  + 0x^{3}   4x^{2}  + 8x   3 
3x^{4}  4x^{2} + 8x  3 
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