Page 222 - DMTH403_ABSTRACT_ALGEBRA

Unit 22: Polynomial Rings

Thus, an example of an element from Z [x] is f(x) = 2 x + 3 x + i. Notes
2
4
Here deg f(x) = 2, and the leading coefficient of f(x) is 2 .
Now, for ring R, we would like to see if you can define operations on the set R [x] so that it
becomes a ring. For this purpose we define the operations of addition and multiplication of
polynomials.
Definition: Let f(x) = a,, + a1x + .. + a, x and g (x) = b + b, x + .. + b x be two polynomials in
m
n
0
m
R[X]. Let us assume that m 2 n. Then their sum f(x) + g(x) is given by
f(x) + g(x) = (a,, + b ) + (a, + b )x + .. + (a + b,) x + b x .. + b x .
n+l
m
1
n
0
n+1
n
m
For example, consider the two polynomials p(x), q(x).in Z[x] given by
p(x) = 1 + 2x + 3x , q(x) = 4 + 5x + 7x 3
2
Then
p(x) + q(x) = (1+4) + (2+5)x + (3+0) x2 + 7x = 5 + 7x + 3x + 7x .
3
3
2
Note that p (x) + q (x)  Z [X] and that
From the definition given above, it seems that deg (f(x)+g(x)) = max (deg f (x), deg g (x)). But this
is not always the case. For example, consider p(x) = 1 + x and q (x) = 2 + 3x – x in Z [X].
2
2
Then p(x) + q(x) = (1+2) + (0+3)x + (1-1)x = 3 a 3x.
2
Here deg (p(x) + q (x)) = 1 < max (deg p(x), deg q(x)).
So, what we can say is that
deg (f(x) + g(x))  max (deg f(x), deg g(x))
 f(x), g(x)  R[x].
Now let us define the product of polynomials.
Definition: If f(x) = a,, + a x + .. + a, x and g(x) = b + b, x + .. + b x are two polynomials in R [x],
m
n
1
m
0
we define their product f(x). g(x) by
f(x) . g(x) = c a c x +.. + c m+n x m+n
0
1
where c = a b , + a b + .... a b  i = 0,l ,... ; m + n.
0 i
i-1 1
1 0
1
Note that a = 0 for i > n and b = 0 for i > m,
i
i
As an illustration, let us multiply the following polynomials in Z[x] :
p(x) = 1 – x + 2x , q(x) = 2 + 5x + 7x .
3
2
Here a, = 1, a, = –1, a = 0, a = 2, b = 2, b, = 5, b = 7.
2
2
3
0
5
i
Thus, p(x) q(x) =  c x , where
i
i=0
c = a b = 2,
0
0 0
c = a b + a b = 3,
1
1 0
0 1
c = a b + a b + a b = 2,
0 2
2 0
1
l
2
c = a b + a b + a b + a b = –3 (since b = 0).
3 3 0 2 1 1 2 0 3 3
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