Polynomials are algebraic expressions that contain terms that are constructed from variables and constants. Recall the rules for adding and subtracting algebraic expressions, which state that only like terms can be combined. 

Like terms are those that are either both constants or have the same variables with the same exponents. For example, $4x^3$ and $x^3$are like terms; $21$ and $82$ are also like terms. Adding and subtracting polynomials is as simple as adding and subtracting like terms. When adding polynomials, the commutative property allows us to rearrange the terms to group like terms together. 

Note that any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials. The resulting polynomial will have the same degree as the polynomial with the higher degree in the problem.

You may be asked to add or subtract polynomials that have terms of different degrees. For example, one polynomial may have the term $x^2$, while the other polynomial has no like term. If any term does not have a like term in the other polynomial, it does not need to be combined with any other term. It is simply carried down, with addition or subtraction applied appropriately. See the second example below for a demonstration of this concept.

Example 1

Find the sum of $4x^2 - 5x + 1$ and $3x^2 - 8x - 9$.

First, group like terms together:

$(4x^2 +3x^2 ) + (- 5x-8x) + (1 - 9)$

Combine the like terms for the solution:

$7x^2 - 13x - 8$

Example 2

Subtract: $(5x^3 + x^2 + 9) - (4x^2 + 7x -3)$

Start by grouping like terms. Remember to apply subtraction to each term in the second polynomial. Note that the term $5x^3$ in the first polynomial does not have a like term; neither does $7x$ in the second polynomial. These are simply carried down.

$5x^3 + (x^2 - 4x^2) + (- 7x) + (9 - (-3)) \\ 5x^3 + (x^2 - 4x^2) - 7x +(9 + 3)$

Now combine the like terms:

$5x^3 - 3x^2 - 7x + 12$

Notice that the answer is a polynomial of degree 3; this is also the highest degree of a polynomial in the problem.