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 1.8.1A polynomial is an expression of the form a_{n}λ^{n} + a_{n−1}λ^{n−1} +

⋅⋅⋅

+ a_{1}λ + a_{0},a_{n}≠0 where the a_{i}are numbers. Two polynomials are equal means that the coefficients matchfor each power of λ. The degree of a polynomial is the largest power of λ. Thus the degree of theabove polynomial is n. Addition of polynomials is defined in the usual way as ismultiplication of twopolynomials.The leading term in the above polynomial is a_{n}λ^{n}. The coefficient of the leading termis called the leading coefficient.It is called a monicpolynomial 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.

Lemma 1.8.2Let f

(λ)

and g

(λ)

≠0 be polynomials.Then there exist polynomials, q

(λ)

and r

(λ)

suchthat

f (λ) = q(λ)g (λ) + r(λ)

where the degree of r

(λ)

is less than the degree of g

(λ)

or r

(λ)

= 0. These polynomials q

(λ)

and r

(λ )

areunique.

Proof:Suppose that f

(λ)

− q

(λ)

g

(λ)

is never equal to 0 for any q

(λ)

. If it is, then the conclusion
follows. Now suppose

r(λ) = f (λ)− q (λ)g (λ )

and the degree of r

(λ )

is m ≥ n where n is the degree of g

(λ)

. Say the leading term of r

(λ)

is
bλ^{m} while the leading term of g

(λ)

is

ˆ
b

λ^{n}. Then letting a = b∕

ˆ
b

, aλ^{m−n}g

(λ)

has the same
leading term as r

(λ)

. Thus the degree of r_{1}

(λ)

≡ r

(λ )

− aλ^{m−n}g

(λ )

is no more than m − 1.
Then

( )
◜---q1◞(λ◟)---◝
r1(λ) = f (λ)− (q(λ)g(λ)+ aλm− ng(λ)) = f (λ)− |( q(λ)+ aλm −n|) g(λ)

Denote by S the set of polynomials f

(λ)

− g

(λ )

l

(λ)

. Out of all these polynomials, there exists one
which has smallest degree r

(λ)

. Let this take place when l

(λ)

= q

(λ)

. Then by the above argument, the
degree of r

(λ)

is less than the degree of g

(λ)

. Otherwise, there is one which has smaller degree. Thus
f

(λ )

= g

(λ)

q

(λ)

+ r

(λ)

.

As to uniqueness, if you have r

(λ)

,

ˆr

(λ)

,q

(λ)

,

ˆq

(λ)

which work, then you would have

(ˆq(λ)− q(λ))g(λ) = r(λ)− ˆr(λ)

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