In this Algebra video tutorial, we explain how to add and multiply polynomials and give formulas for the degree of a sum or product of polynomials. We also define what coefficient is, and briefly touch on the subject of additive and multiplicative inverses of polynomials.

To begin, we give an example of adding two abstract first degree polynomials. Below, we use the letters c and d with subscripts to represent the constant coefficients of our x and constant terms. Reading from the top left to the bottom right, we apply the associative property to regroup the sum. Then we use commutativity to reorder the terms in the parentheses. Finally, we use associativity and the distributive property to regroup the terms and factor out an x from the first group.

Next, we show demonstrate how to add two concrete polynomials. Like our previous abstract example, both of these polynomials are first degree. Everything is the same as the previous example, except that we use 3 and 5 for the values of c1 and c0 and 2 and 4 for the values d1 and d0.

When we add two polynomial, the degree of the resulting polynomial sum is less than or equal to the higher degree of the two polynomials that we are adding together. The polynomial sum may have lower degree because the monomials with the highest degrees can cancel. When we multiply two polynomials, the resulting polynomial product has its degree equal to the sum of the degrees of the polynomials that we are multiplying, as we will see below

Below, we demonstrate the multiplication of two abstract first degree polynomials. Again, we use the letters c and d with subscripts to represent the constant coefficients of our x and constant terms. In our first step, we use the distributive property to distribute the terms of our first polynomial across the second. Then we use distributivity again to distribute the terms of the second polynomial. Finally, we use distributivity to factor out x from the middle two terms. The result is a second degree polynomial, as expected.

Next, we show how to multiply two concrete polynomials: x - y + 2 and xy - 3. Just as before, we begin by distributing the terms of the first polynomial through the second polynomial. Then we use the distributive property again to distribute the terms of the second polynomial and get six terms. Since none of these terms can be combined, we use commutativity to rearrange them in order of descending degree.

Finally, we mention we can get the additive inverse of a polynomial simply by negating each of its terms; this is equivalent to multiplying the polynomial by -1. The multiplicative inverse is trickier and is almost never a polynomial. However, we will wati to take up the issue or multiplicative inverses at a later time.