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Rings of Polynomials
HW p. 8 # 1-7 at the end of the notes
Definition 1: Ring of Polynomials over R: Let R be a commutative ring. The set of
formal symbols
R[ x]  {an x n  an1 x n1    a1 x  a0 | ai  R, n  Z  }
is called the ring of polynomials over R in the indeterminate form x.
Note: If f ( x)  an x n  an1 x n1    a1 x  a0  R[ x] , where a n  0 , the f has degree
n. If the leading coefficient a n  1 , the identity in R, then f is said to be monic. A
constant polynomial f ( x)  a0 , a0  0 has degree zero. The polynomial f ( x)  0 has no
degree.
For example, the polynomial f ( x)  x 4  x 3  2 x  1 is a monic polynomial of degree 4
in Z 5 [ x] . The polynomial g ( x)  3x 5  x  5 is a non-monic polynomial of degree 5 in
Z 6 [ x] .
Facts about Polynomials
1. Two polynomial elements of the same degree of in R[x] are equal if and only if their
corresponding coefficients are equal. That is, for
f ( x)  an x n  an1 x n1    a1 x  a0
g ( x)  bn x n  bn1 x n1    b1 x  b0
f ( x)  g ( x) if and only if ai  bi for i  0, 1, 2,  n
2. Polynomials in R[x] are added in the usual way, where the coefficients of like terms
are added.
3. Polynomials in R[x] are multiplied in the usual way, using the distributive laws. We
even using the FOIL method on binomials since the ring R is commutative.
4. If R has unity 1, then the unity polynomial for R[x] is f ( x)  1 .
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Example 1: Find the sum and product of the polynomials f ( x)  4 x 3  2 x 2  x  3 and
g ( x)  3x 4  2 x 3  x 2  4 x  1 in Z 5 [ x] .
Solution:
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Theorem 1: If R is an integral domain and if f (x) and g (x ) are non-zero polynomials
in R[x] , then
deg[ f ( x)  g ( x)]  deg f ( x)  deg( g ( x))
Proof: Suppose f ( x)  an x n  an1 x n1    a1 x  a0 is a polynomial of degree n and
g ( x)  bm x m  bm1 x m1    b1 x  b0 be a polynomial of degree m. Then the leading
term for f ( x)  g ( x) is
a n bm x n m .
Since a n  0 and bm  0 , then an bm  0 since R is an integral domain. Hence,
f ( x)  g ( x) is nonzero and deg[ f ( x)  g ( x)]  n  m  deg f ( x)  deg g ( x) .
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For two integers a and b where a  b , the division algorithm says there exists unique
integers q and r were when computing a  b , a  qb  r where 0  r  b . There is a
similar statement for polynomials.
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Theorem 2: Division Algorithm for F [x ] : Let F be a field and let f ( x), g ( x)  F [ x]
with g ( x)  0 . Then there exists unique polynomials q( x), r ( x)  F [ x] where
f ( x)  q( x) g ( x)  r ( x) (when taking f ( x )  g ( x ) ) where either r ( x)  0 or
deg r ( x)  deg g ( x) .
Proof: We first show the existence of q(x) and r (x ) . If f ( x)  0 or
deg f ( x)  deg g ( x) ,then set q( x)  0 and r ( x)  f ( x) , which gives
f ( x)  0  g ( x)  f ( x)  f ( x) . So we may assume that
n  deg f ( x)  deg g ( x)  m
Let f ( x)  an x n  an1 x n1    a1 x  a0 and g ( x)  bm x m  bm1 x m1    b1 x  b0 .
We will use induction to prove the existence of q(x) and r (x ) .
Case 1: Assume deg f ( x)  0 . Then deg g ( x)  0 and f ( x)  a0 and g ( x)  b0 . Since
F is a field, every element has a multiplicative inverse and we write
f ( x)  a0 (1)  a0 (b01b0 )  0  (a0b01 )b0  0  (a0b01 ) g ( x)  0 .
Hence, we set q( x)  a0 b01 and r ( x)  0 .
Case 2: Assume that the division algorithm holds for all polynomials of degree less than
the degree f ( x)  n . We form the polynomial
f1 ( x )  f ( x ) 
an x n
bm x m
g ( x)
 a n x  a n1 x
n 1
 a n x  a n1 x
n 1
n
n
   a1 x  a0 
   a1 x  a0 
an x n
bm x
m
an x n
bm x
b
mx
m
an x n
bm x
m
b
m 1 x
 bm1 x m1    b1 x  b0
bm x 
m
 a n x n  a n1 x n1    a1 x  a0  a n x n 
 a n 1 x n1    a1 x  a 0 
m
an x n
bm x
m 1
m
an x n
bm x
b
m
m 1 x
b
m 1 x
m 1
   b1 x  b0
m 1
   b1 x  b0
   b1 x  b0

Hence, n  1  deg f1 ( x)  deg f ( x)  n .
By the induction hypothesis, there exists polynomials q1 ( x) , r1 ( x) where
f1 ( x)  q1 ( x) g ( x)  r1 ( x) , 0  deg r1 ( x)  deg g ( x) .



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Solving for f (x) gives
f ( x )  f1 ( x ) 
an x n
bm x m
g ( x)
 q1 ( x) g ( x)  r1 ( x) 

an x n

 q1 ( x) 

bm x m

an x n
bm x m
g ( x)

 g ( x)  r1 ( x)


Thus, setting q ( x)  q1 ( x) 
an x n
bm x m
and r ( x)  r1 ( x) proves the existence of q(x) and
r (x ) .
To show the uniqueness of q(x) and r (x ) , suppose
f ( x)  q1 ( x) g ( x)  r1 ( x) where deg r1 ( x)  deg g ( x) or deg r1 ( x)  0
f ( x)  q2 ( x) g ( x)  r2 ( x) where deg r2 ( x)  degree g ( x) or deg r2 ( x)  0
We want to show that q1 ( x)  q2 ( x) and r1 ( x)  r2 ( x) . If we take
f ( x)  q1 ( x) g ( x)  r1 ( x)
f ( x)  q2 ( x) g ( x)  r2 ( x)
and subtract the two equations we get
0  q1 ( x)  q2 ( x)g ( x)  r1 ( x)  r2 ( x)
or
r2 ( x)  r1 ( x)  q1 ( x)  q2 ( x)g ( x)
Now
degr2 ( x)  r1 ( x)  deg( g ( x))  degq1 ( x)  q2 ( x)g ( x)
This can only happen if r2 ( x)  r1 ( x)  0 or r2 ( x)  r1 ( x) . Thus q1 ( x)  q2 ( x)  0
or q1 ( x)  q2 ( x) . Hence, we are done.
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Example 2: Find the quotient and remainder upon division of the polynomials
f ( x)  x 4  x 2  7 x  2 and g ( x)  3x 3  x 2  18x  9 in Q[x ] .
Solution:
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Example 3: Find the quotient and remainder upon division of the polynomials
f ( x)  x 4  x 3  2 x 2  2 and g ( x)  3x 2  4 x  2 in Z 5 [ x] .
Solution:
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Exercises
1. Find the sum and product of the polynomials for the given polynomial ring.
a. f ( x)  4 x  5, g ( x)  2 x 2  4 x  2 in Z 8 [ x] .
b. f ( x)  x  1, g ( x)  x  1 in Z 2 [ x] .
c.
f ( x)  2 x 2  3x  4, g ( x)  3x 2  2 x  3 in Z 6 [ x] .
d.
f ( x)  2 x 3  4 x 2  3x  2, g ( x)  3x 4  2 x  4 in Z 5 [ x] .
2. Find the quotient and remainder upon division of the polynomials the given
polynomial ring.
a. f ( x)  x 6  3x 5  4 x 2  3x  2, g ( x)  x 2  2 x  3 in Q[x ] .
b.
f ( x)  x 6  3x 5  4 x 2  3x  2, g ( x)  x 2  2 x  3 in Z 7 [ x] .
c.
f ( x)  x 6  3x 5  4 x 2  3x  2, g ( x)  3x 2  2 x  3 in Z 7 [ x] .
d.
f ( x)  x 5  2x 4  3x  5, g ( x)  2x  1 in Q[x ] .
e.
f ( x)  x 5  2 x 4  3x  5, g ( x)  2 x  1 in Z11[ x] .
f.
f ( x)  x 4  5x 3  3x 2 , g ( x)  5x 2  x  2 in Z 5 [ x] .
3. Prove that if R is an integral domain, then R[x] is an integral domain.
4. Including the zero polynomial, how many polynomials are there of degree 3 in Z 2 [ x] ?
How many of those polynomials are monic?
5. Including the zero polynomial, how many polynomials are there of degree equal to 2 in
Z 5 [ x] ? How many of those polynomials are monic?
6. Including the zero polynomial, how many polynomials are there of degree less than or
equal to 3 in Z 2 [ x] ? How many of those polynomials are monic?
7. Including the zero polynomial, how many polynomials are there of degree less than or
equal to 2 in Z 5 [ x] ? How many of those polynomials are monic?
8. If F is a field, show that F [x ] is not a field. (Hint: Is x a unit in F [x] ?)
9. Let R be an integral domain. Assume that the Division Algorithm always holds in R[x]
Prove that R is a field.