1.5. The Field of Algebraic Expressions
Our brief excursion into the world of polynomials led us to the following observations.
Operations on real numbers carry over to operations on polynomials because polynomials are
merely formulas for computing some real numbers. The parallelism, however, ends when we
encounter division. Division of polynomials does not always yield a polynomial. The
multiplicative inverse of a polynomial is not necessarily a polynomial. The most that we can say
about the structure of polynomials is that it is a commutative ring with identity. There is a need
for a structure where the multiplicative inverse of a polynomial is legitimate.
Be ready for the
field of algebraic expressions!
The Field of Quotients
Let P(x) be a polynomial of the form
a
0
+ a
1
x + a
2
x
2
+ a
3
x
3
+ . . . + a
n
x
n
.
Consider the set of algebraic expressions which can be written as quotients of polynomials,
which will be called
rational expressions
. Thus, a rational expression is of the form P(x)/R(x),
where R(x) ≠ 0. For example,
5
and
,
1
x
,
x
1
,
1
x
1
x
2
x
3
2
are rational expressions, while the following are not:
2
3
2
x
1
x
2
x
and
x
1
x
,
x
,
x
(Why?)
We shall agree to write each rational expression in simplest form, i.e., each rational expression
can be written as a quotient of polynomials which are relatively prime or with no common factor
except 1. To this end, we shall use the algebraic fact that
,
b
a
c
b
c
a
provided b, c ≠ 0.
Similar rational
expressions are those that have the same denominators. Thus
2
2
x
1
x
and
x
2
are similar while
3
x
4
and
x
3
are not. However, dissimilar rational
expressions can be made similar by using, again, the algebraic fact stated above.
Time to think!
Express the following in simplest form. What conditions shall you impose?
2
2
2
2
3
2
2
y
x
y
x
)
3
3
x
27
x
)
2
10
x
3
x
35
x
12
x
)
1
Time to think!
Write the following as similar rational expressions:
x
2
1
,
2
x
2
)
4
)
1
x
(
x
,
1
x
1
)
3
3
x
1
,
2
x
x
)
2
1
x
2
,
1
x
3
)
1
3