Exercise 2.92: By imposing an ordering on
variables, extend the polynomial package so that addition and multiplication of
polynomials works for polynomials in different variables. (This is not easy!)
Extended exercise: Rational functions
We can extend our generic arithmetic system to include
rational functions.
These are “fractions” whose numerator and denominator are
polynomials, such as
x
+
1
x
3
−
1
.
The system should be able to add, subtract, multiply, and divide rational
functions, and to perform such computations as
x
+
1
x
3
−
1
+
x
x
2
−
1
=
x
3
+
2
x
2
+
3
x
+
1
x
4
+
x
3
−
x
−
1
.
(Here the sum has been simplified by removing common factors. Ordinary “cross
multiplication” would have produced a fourth-degree polynomial over a
fifth-degree polynomial.)
If we modify our rational-arithmetic package so that it uses generic
operations, then it will do what we want, except for the problem of reducing
fractions to lowest terms.
练习 2.92：通过对变量施加排序，扩展多项式包，使多项式的加法和乘法能够适用于不同变量的多项式。（这并不容易！）
扩展练习：有理函数
我们可以扩展通用算术系统，纳入有理函数 (rational functions)。有理函数是以多项式为分子和分母的"分数"，例如
(x + 1) / (x³ − 1)。
系统应能对有理函数进行加、减、乘、除，并能完成如下计算：
(x + 1) / (x³ − 1) + x / (x² − 1) = (x³ + 2x² + 3x + 1) / (x⁴ + x³ − x − 1)。
（这里的和式已通过约去公因子化简。普通的"交叉相乘"会得到一个五次多项式作分母的四次多项式。）
如果我们修改有理算术包使其使用通用操作，那么除了将分数化为最简形式这一问题之外，其他一切都能如期运作。