ADDING THE LITTLE BUGGERS:

To begin our discussion of adding polynomials, we will need to clear up a bit of terminology.

First we have little things known in the math world as:

monomials: these are single numbers, single variables, or products of numbers and variables. Here are some examples of monomials:
2
-13
3.5
x
1/2x
xy
-8xy^2

binomials: as the prefix "bi" suggests, a binomial is TWO monomials added together. Here are a few examples of binomials:
2+x
-12y+(-3) ......more commonly written (-12y - 3)
x+y
3x^2+2y

trinomials: once again the prefix helps here. A trinomial is a sum of THREE monomials. Here are some examples of trinomials:
8x^2+2x+1
-5z^2+(-4z)+(-2) ..more commonly written (-5z^2-4z-2)
6r^3y+19rs+9s^3

polynomials: now this is the general word for any monomial or any sum of monomials. Remember, a polynomial may have any number on monomials (also known as terms) in it.

constant term: is a monomial that is just a number, no variable involved. These are traditionally placed at the end of the polynomials.

coefficient: the coeficient of a monomial is the number part, not the variable part. Coefficients are placed in the front of the monomial.
ex: 3x........ three is the coefficient
ex: -15.6y^2xz........ -15.6 is the coefficient here
ex: x.......there is a coefficient of one here, even though you don't see it. We say it is an "understood" one.
ex: -y......there is an "understood" coefficient of -1 here.

degree of a monomial: is the sum of the exponents of the variables. If there is no exponent, the degree is 0.
ex: The degree of 3x is 1.
ex: The degree of 5x^2 is 2.
ex: The degree of 2x^3y^2 is 5.
ex: The degree the constant term 4 is 0.

degree of a polynomial: To find the degree of a polynomial, just look at each monomial in it, and select the monomial with the highest degree. This will be the degree of the entire polynomial.

ex: The degree of 9x^3y^4 + 3xy -7 is 7.

descending order: this means that the polynomial is written with the term with the highest power of the variable listed first on the left. Then each lesser power term is listed in descending order.
ex: 3x^3 + -4x^2 - 9x +1
ex: y^5 + 2y^3 - 8 .....notice here it appears that the fourth, second and first power terms are missing.....we say they have a coefficient of zero.
technically we could write the above polynomial as:
y^5 + 0y^4 + 2y^3 + 0y^2 + 0y - 8

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With the above all said, now let's go ahead and start to add polynomials.

The absolute most important thing you must remember when doing this is that you can only add monomials that are the SAME SHAPE. We say they are "like terms".

This may seem a bit strange, but remember our Algebra Tiles. Yes many algebraic expressions can be visually represented. Why don't you hop on over to the Adding Like Terms with Algebra Tiles demonstration to see them in action.

As long as you remember that to be "like terms" the variable portion of the monomials must match EXACTLY, you will be ready to start to add.

Let's give it a shot:
ADD:
6x+9
x+(-1).......click for answer

ADD:
(8x^2+2x+1) + (x^2-4x+7)........click for answer

ADD:
-5x^2-5x+3
-13x^2+9x-14......click for answer

ADD:
( x^3y+3x^2y^2+2xy^3)
(2x^3y-9x^2y^2-xy^3)
....click for answer
*Remember, in problems such as this, there is an "understood" 1 in front of any variable that appears not to have a coefficient. This is why the answer starts with 3x^3y (we added the understood coefficient of 1 to the other coefficient of 2 and got 3x^3y).