It will be very important to be able to work with polynomials in certain parts of linear algebra to be presented later. It is surprising how useful this junior high material will be.
Definition 1.11.1 A polynomial is an expression of the form

a_{n}≠0 where the a_{i} are numbers. Two polynomials are equal means that the coefficients match for each power of λ. The degree of a polynomial is the largest power of λ. Thus the degree of the above polynomial is n. Addition of polynomials is defined in the usual way as is multiplication of two polynomials. The leading term in the above polynomial is a_{n}λ^{n}. The coefficient of the leading term is called the leading coefficient. It is called a monic polynomial if a_{n} = 1.
Note that the degree of the zero polynomial is not defined in the above. The following is called the division algorithm.

where the degree of r
Proof: Suppose that f

and the degree of r

Denote by S the set of polynomials f
As to uniqueness, if you have r

Now if the polynomial on the right is not zero, then neither is the one on the left. Hence this would involve two polynomials which are equal although their degrees are different. This is impossible. Hence r