Polynomials are multiplied by using the . For example, for the polynomial product
it is possible to distribute to every term of Then, by using the Distributive Property once more, the product can be expressed explicitly.
When multiplying two polynomials, each term of the first polynomial multiplies with each term of the second one. This have some important consequences:
- The product of a polynomial with terms and a polynomial with terms results in products. For example, the product of two gives products as result.
- Since each product of terms results in a new , the result of a polynomial multiplication is a sum of monomials, which is, by definition, another polynomial. Therefore, polynomials are closed under multiplication.
- When two polynomials are multiplied, the product is a new polynomial with a that is the sum of the degrees of the factor polynomials. This follows from the when multiplying the highest of the factor polynomials.
Find the product of the polynomials and by distributing to each term in Then, determine the degree of the resulting polynomial.
It can be seen that since both polynomials have terms, multiplying them results in products. Nevertheless, it can be simplified by
It can also be noted that the polynomial its of degree and the polynomial its of degree When written in , their product is
This is a polynomial of degree
In order to make the multiplication process systematic there are different methods that can be used, such as the and using tables of products. Ultimately, all of these methods are based on the Distributive Property. An example of a table of products is shown below.